1

i

SAIL THROUGH MATHEMATICS:STRUCTURED ACTIVITIES FORINTELLIGENT LEARNING

The author Richard Skemp is internationally recognized as a pioneer in the psychol-ogy of learning mathematics based on understanding rather than memorizing rules. A former mathematics teacher, he has lectured in twenty countries, is a former Pro-fessor of Educational Theory and Director of the Mathematics Education Research Centre at Warwick University, and a Past President of the International Group for the Psychology of Mathematics Education.

ii

Original (1989) Edition:

Published in England as Structured Activities for Primary Mathematics, Volume 2 Part of Routledge Education Books Advisory Editor: John Eggleston Professor of Education University of Warwick

Expanded (1994) North American Edition:

Published in Canada as SAIL through Mathematics, Volume 2 Edited and Adapted by:

Bruce Harrison Professor Emeritus of Curriculum and Instruction (Mathematics Education) The University of Calgary and EEC Ltd.

Marilyn Harrison Teacher and Mathematics Education Consultant Calgary Board of Education and EEC Ltd.

iii

SAIL THROUGH MATHEMATICS: STRUCTURED ACTIVITIES FOR INTELLIGENT LEARNING

Richard R. Skemp Emeritus Professor, University of Warwick

Volume 2

EEC Ltd. Calgary

iv

First published in 1989 byRoutledge11 New Fetter Lane, London EC4P 4EE

© Richard R. Skemp 1989, 1994

Expanded North American Editionedited and adapted byBruce and Marilyn Harrisonwith permission from Routledge.

All rights reserved. No part of this book may bereprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, nowknown or hereafter invented, including photocopyingand recording, or in any information storage orretrieval system, without permission in writing from the publishers.

As suggested on page 5, permission is hereby granted to reproduce, non-commercially , relevant sections of the activity instructions for inclusion with made-up activities.

British Library Cataloguing in Publication Data

Skemp, Richard R. (Richard Rowland), 1919- Structured activities for primary mathematics: how to enjoy real maths.– (Routledge education books). Vol.II 1. Primary schools. Curriculum subjects: Mathematics I. Title 372.7’3 British ISBN 0-415-02819-1

Expanded North American Editionpublished in 1994 byEEC Ltd.6016 Dalford Road, N.W., Calgary T3A 1L2

ISBN 0-9697190-2-7

Printed in Canada byMountain View Printing and Graphics Ltd.Calgary

v

CONTENTS Acknowledgements vi Notes vii

A Teacher’s Guide 1 Classroom management 1 Suggested sequencing 2 Materials 3 Data management 6 Evaluation 6 Meeting individual needs 7 Across the curriculum 8 Problem solving 8 Meeting the NCTM Standards 9 The Standards and Professional Development 11

Introduction 13 1 Why this book was needed and what it provides 13 2 The invisible components and how to perceive them 14 3 How this book is organized and how to use it 17 4 Getting started as a school 19 5 Organization within the school 20 6 Getting started as an individual teacher 21 7 Parents 22 8 Some questions and answers 23

Concept Maps and Lists of Activities 27

THE NETWORKS AND ACTIVITIES 65activity codes [Org 1] Set-based organization 67 [Num 1] Numbers and their properties 69 [Num 2] The naming of numbers 85 [Num 3] Addition 113 [Num 4] Subtraction 125 [Num 5] Multiplication 151 [Num 6] Division 187 [Num 7] Fractions 219 [Space 1] Shape 263 [Space 2] Symmetry and motion geometry 313 [NuSp 1] The number track and the number line 359 [Patt 1] Patterns 381 [Meas 1] Length 385 [Meas 2] Area 405 [Meas 3] Volume and capacity 433 [Meas 4] Mass and weight 437 [Meas 5] Time 449 [Meas 6] Temperature 457

Glossary 463 Alphabetical List of Activities 465 Sequencing Guides 469 Progress Record 479

vi

ACkNOwLEDGEMENTSI am very grateful to all the following for their help. TheNuffieldFoundationandtheLeverhulmeTrust,fortheirfinancialsupport of the Primary Mathematics Project during a total period of eight years. Mr David Pimm and Ms Janet Ainley, Research Associates during phases one and two, and my wife Valerie, Project Assistant during phase three, for their contri-butions to so many aspects of this project. (David Pimm is Lecturer in Mathemati-cal Education at the Open University, and Janet Ainley is now Lecturer in Primary Mathematics at Warwick University.) In the Solihull Local Education Authority: Mr Colin Humphrey, Director of Education, the late Mr Paul Turner and his successor Mrs Marion Idle, Mathemat-ics Advisers, Mr Alan Stocks, Assistant Director of Education, Mrs Barbara Fur-niss, Head Teacher, and the staff and children of Bentley Heath Primary School; Mrs Rita Chapman, Head Teacher and the staff and children of Lady Katherine Leveson’s Primary School. In the Dyfed Local Education Authority, Dr David Finney, former Mathematics Adviser, Mrs A. Cole, Head Teacher, and the staff and children of Loveston County Primary School. In the Shropshire Local Education Authority: Dr David Finney, Science Adviser; Mrs Susan Boughey, Head Teacher, and the staff and children of Leegomery County Infant School; Mrs Ishbel Gam-ble, Head Teacher, and the staff and children of Holmer Lake County First School; Mr David Tyrell, Head Teacher and the staff and children of Leegomery County Junior School; Mrs Pamela Haile, Head Teacher, and the staff and children of St Lawrence Primary School. In the Warwickshire Local Authority: Mrs Wyatt, Head Teacher, and the staff and children of Haselor Primary School; Mrs Anne Dyson, Head Teacher, and the staff and children of Wellesbourne Primary School. I am grateful to all who allowed me to work in their schools during the pilot testing of the project materials, and also for their advice, suggestions, and helpful discus-sions; and to the children for the many hours of enjoyment which I have had while working with them. My colleague Professor John Eggleston, Professor of Education at Warwick University, and the Advisory Editor for Routledge Education Books, for his en-couragement and editorial advice. In Alberta, Canada: Merlin Leyshon, Principal, Marilyn Harrison, and the teachers at R. B. Bennett Elementary School, Calgary, who pioneered Canadian whole-schoolfieldtestingoftheactivitiesformorethanfiveyears;LynnFossum,Principal, Sandra Carl, and teachers at Oliver School, Edmonton; Rick Johnson, Edmonton Public School Board consultant; Peter Murdock, Principal, and students fromTheWaldorfSchoolinCalgary,whohelpedfieldtesttheAreaMeasurementactivities; Grant Spence, Principal, Val Eckstrand, and the pupils at Ross Ford Elementary School, Didsbury, who have been implementing the SAIL program and graciously agreed to have photographs taken for SAIL, Volume 2. The more than 700 Alberta teachers who have so enthusiastically attended inservice courses on the learning activities found in these volumes and who have been so generous in their comments. Andfinally,butbynomeansleast,yourselves,forallowingoureffortstofindtheir hoped-for destination in your own classrooms.

vii

NOTES

SAIL through Mathematics, Volume 1 is intended for Pre-grade Level 1 through Grade Level 2, and Volume 2 for Grade Levels 3 through 6. Because of the wide range of children’s abilities, this division can only be very approximate. In par-ticular, some of the later activities in Volume 1 will be found useful for children in Grade Level 3, especially if they come from schools where this approach has not been used. Every administrative grade placement brings together children cur-rently operating at many different developmental levels in mathematics. The SAIL program is uniquely designed to accommodate a wide range of abilities. Since the English language lacks a pronoun which means either he or she, I have used these alternately by topics.

viii

SAIL Activities in the Classroom

1

Teacher’s Guide

A TeAcher’s Guide to structured Activities for intelligent Learning

Drawing on ten years of experience using Richard Skemp’s structured learning activities with her own classes and with many other teachers and their students, Marilyn Harrison has written the following teacher’s guide and the accompanying sequencing guides to share insights, ideas, and suggestions.

setting sail

You are about to embark on an adventure in learning mathematics with understand-ing. The vehicle for this adventure consists of a set of structured and sequenced mathematics learning activities that have been designed in accordance with contem-porary learning theory. Cooperative, schematic learning and prediction from math-ematical patterns provide the basis for a powerful and unique approach to problem solving. Clusters of activities enable students to construct networks of mathematical ideas intelligently by building and testing through direct experience, communication with others, and creative extrapolation.

The learning activities in the SAIL volumes provide opportunities for cooperative group work, discussion, conjecturing, analyzing, justifying, writing, exploring, and applying mathematics. Where appropriate, memorization is encouraged for fluency and efficiency, but never in place of developing reasoning and understanding. The activities are designed for students to learn effectively by doing, saying, and record-ing. They explore a variety of appropriate methods. Prediction activities occur frequently, providing an excellent context for practicing estimation and reasonable conjecture skills while learning to reason mathematically.

The use of concrete materials in the SAIL program is extensive and carefully de-signed. Exemplary techniques are modelled for drawing the most out of the concrete and the more abstract learning situations.

suggestions for smooth sailing

classroom management The activities are designed to engage students in pairs, in small groups of up to six

students, or in whole class discussions. The optimal group size is indicated in the in-structions for each activity. If students are not accustomed to working cooperatively in mathematics, their social learning skills need to be considered. They may have to learn such things as listening to each other, taking turns, discussing sensibly, and giving reasons rather than just arguing. The ways of learning mathematics which are embodied in SAIL both depend upon, and contribute to, social learning and clear speech. If these traits are already well established, you are off to a flying start.

2

Teacher’s guide

introducing the activities It is the responsibility of a teacher to be familiar with each activity and its purpose

before introducing it to students. To introduce an activity to the whole class, gather students in a circle and demonstrate the procedures by leading the activity with one or two students. Then have them do the activity in appropriate groups. Tell them to check that all the materials are on hand before they begin and that all the materials go back in their proper places at the end. It is worth the effort to establish well-defined routines from the beginning and most students appreciate the importance of keeping everything in place for those who will next do this activity. As the activity period concludes, it is important that the teacher and the students together reflect on and discuss their discoveries.

An alternative to introducing a new activity to the whole class would be to initi-ate it with a small group of students while the others are engaged in activities which they have already learned. This provides an opportunity to give ‘high quality input’ to small groups. One of the advantages of organizing your classroom this way is that the other students are learning on their own, doing their own thinking. This kind of teaching includes ways of managing students’ learning experiences which are less direct, more sophisticated, and more powerful than traditional approaches.

You will find that the activities fall into two main groups: those which introduce new concepts, and those which consolidate them and provide a variety of applica-tions. Activities in the first group always need to be introduced by a teacher to ensure that the right concepts are learned. Once they have understood the concepts, students can go on to do the consolidating activities together with relatively little supervision. These activities could be introduced by another adult helper. In some cases students, especially older students, can teach it to others with the help of the printed rules.

A set of 39 videoclips of the SAIL activities and of the theory which they em-

body has been produced. Each of the 3- to 11-minute activity videoclips models the introduction of a SAIL activity to a small group of children. Five of the videoclips demonstrate classroom management skills when the activities are used with a whole class. A 60-minute Discussion Time with Richard Skemp video provides a compre-hensive overview of his theory of intelligent learning, illustrated with sample learn-ing activities.

charting the course

suggested sequencing To help sequence the activities, especially for teachers new to the program, sug-

gested grade-by-grade sequencing guides covering Grade Level 3 through Grade Level 6 have been included in the final pages of SAIL Volume 2. The Grade Level 3 sequencing includes concepts appropriate for most 8-year-olds; Grade Level 4, for most 9-year-olds; Grade Level 5, for most 10-year-olds; and Grade Level 6, for most 11-year-olds. There are many possible routes through the Networks. The suggested sequencing, though not unique, is offered as an aid for those wishing to be assured that the curricular expectations of each level are covered.

3

Teacher’s Guide Materials Many of the materials are found in most schools. The following list will enable you

to quickly check which materials may be needed.

Collectables Enlist the help of children and parents to provide the following:

• thin cord or yarn for set-loops

• little objects for sorting (natural objects – we do not want children to think that mathematics only involves plastic cubes): buttons, sea shells, beans, peb-bles, bottle tops, keys, pasta, bread tags, nuts and bolts, seeds, crayons, screws (store in small containers, boxes or see-through plastic bags)

• sorting trays – trays with partitions; egg cartons; box lids

• catalogues and magazines for assembling pictures

• shaker for dice (though rolling the die without a shaker is fine)

• containers: egg cups (or 35 mm film containers); identical drinking glasses; an assortment of glasses, jugs, jars, varying as widely as possible in height and width; two identical containers of capacity equal to or greater than any of the preceding

• a collection of objects which approximate in shape to spheres, cuboids, cylin-ders, and cones; and others which are none of these

Purchased materials

• interlocking cubes – 1 cm (e.g., Centicubes)

• interlocking cubes – 2 cm (e.g., Multilink cubes) Caution: Some commer-cially available linking cubes have dimensions slightly smaller than 2 cm and may cause problems when used with the 2 cm number tracks provided in the photomasters, especially when ‘rods’ are made and checked against a number track. If the dimensions of available cubes are not exactly 2 cm, alternate number tracks with appropriate dimensions could be drawn to avert unneces-sary confusion for the children.

• attribute blocks

• ‘1-6’ dice

• ‘0-9’ dice

• dice bearing only 1’s and 2’s (e.g., cover the faces with sticky paper and write

‘1’ on half of them, ‘2’ on the other half)

• 1¢, 5¢, 10¢, 25¢, and $1.00 coins

4

Teacher’s guide

• 2, 5, 10, 20, 50 dollar bills (play money)

• milk straws and popsicle sticks

• base ten material – ones, tens, hundreds, thousands

• thousands, hundreds, tens, and ones charts

• geometric models of spheres, cuboids, cubes, cylinders, cones, pyramids &

prisms (a set of geometric solids)

• Plasticine

• non-permanent markers, i.e., water soluble pens such as those used with over-

head projector acetate film

• coloured felt-tip pens

• squared paper

• rulers marked in centimetres, millimetres

• a metre-stick

• a ‘skeleton’ metre cube

• paper clips – small, medium, large (1 box of each)

• 6 wooden rods 20 cm long; 6 wooden rods 24 cm long

• 3 model trucks (could be made from Lego in various widths and heights)

• resealable plastic bags (e.g., Ziploc freezer bags)

• waxed string (e.g., Wikki Stix) (see Meas 2.2/3)

• balance scales (pan balance, spring balance)

• bolts (see Meas 4.4/1)

• analogue clock face with hour hand

• analogue clock face with hour and minute hands

• Celsius thermometers

• calendars

• calculators

• safety compasses

• protractors

• pocket- or purse-sized mirror or a MIRA

• set-square (i.e., a strudy right-angled triangle, plastic or wooden)

• a magnetic compass

• a set of standard masses (varying from 10 gm to 1 kg)

• a set of plastic beakers (ranging in capacity from 10 mL to 2 L)

5

Teacher’s Guide Preparing the activity cards Two routes are available. You can duplicate the cards from the photomasters in

Volumes 1a or 2a or purchase sets of prepared cards. To keep the card preparation manageable, it is recommended that you work on one Network or one Level at a time.

Perhaps the best way to keep all the materials for each activity together is to store them in a labelled plastic bag or a cardboard box. Plastic bags can be stored upright in a suitably sized open cardboard box. Alternatively, the bags can be suspended from pegboards with hooks. Some teachers find it useful to have with each activity a copy of the relevant ‘Concept,’ ‘Ability,’ ‘Materials,’ and ‘What they do’ sections of the activity instructions. These may be mounted on coloured card with the relevant ‘discussion boxes’ included on the back. The instruction activity cards could be colour coded by Network.

i) Preparing the materials from the Photomasters:

The preparation of materials from the Photomasters is described in Volume 2a on page x. Labelling the bags and underlining in red the materials (such as base 10 materials) which are not kept in these individual bags indicates what else has to be collected at the start of an activity.

ii) Sets of printed card are available either laminated or not.

Not laminated Most of the cards are ready to be laminated but a few need to be taped together

before laminating. These are clearly indicated. Laminate the cards on both sides to prevent curling.

Laminated card Cut out individual cards, place elastic bands around decks of cards, and store

them with the activity boards (if applicable) in plastic bags or boxes. Self- sticking printed labels are provided.

The prepared materials can be stored effectively in a central location or in indi-vidual classrooms. Even when school sets are available, teachers prefer to have cop-ies of some of the consolidation activities in their classrooms. Extra elastic bands should be available in each classroom.

Parent involvement in the preparation of activities is discussed in detail on page 22.

It is worth remembering that, except for occasional replacement, the work of preparing the materials will not have to be repeated. The time spent is a capital investment, which will pay dividends in years to come. You are also contributing to children’s long-term learning which makes all the hard work worth the effort.

6

Teacher’s guide

Managing the Fleet data management Many opportunities arise in classroom settings which spontaneously lead to data

management activities.

Ask children to contribute collectable materials. As the materials are collected record the type of material contributed by each child on a class graph. This will likely encourage them to bring more.

As they complete the activities, ask students to graph their results. Here are some examples.

Renovating a house (Num 3.9/3) Each ‘couple’ keeps a record on a graph of how many months they need to save to

buy each item to renovate their house. Compare with other groups to find out which ‘couple’ is able to do their renovations in the shortest amount of time.

Products practice (Num 5.6/5) Players record their time (after correction of errors) on a graph. Each student could

keep an individual record over a period of time.

evaluation

“In a book about teaching, the importance of assessment is that we must know how far children have reached in their understanding, to know what they are ready to learn next, or whether review and/or consolidation are needed before going on.”1

i) Teacher Observation Observe students as they do the activities. Even when introducing the activities

to a whole class, a teacher has the opportunity to observe individual students as the procedures are demonstrated. This is a valuable opportunity to focus on a few children followed by more as all of the children work in small groups. Notes can be assembled for individual students in the group by making brief comments on ‘post notes’ which can later be filed.

ii) Evaluation checklists – keep a record of individual student progress using the

Progress Record provided at the end of this book.

iii) Self/Peer Evaluation – discussions at the conclusion of each activity period will provide opportunities for individual and peer evaluation. Student journals can provide diagnostic assessment; e.g., “What I already know about division” before introducing the topic, followed by “What I learned about division” after completing the appropriate activities in the division network.

7

Teacher’s Guide iv) Portfolios of students’ work – as students record their work, keep their sheets in

a folder or provide a booklet. Keep a record of the problems they write, the pat-terns they make, etc.

v) Teacher-Child Interview/Conferences – when others are engaged in consolida-tion activities, teachers can have conferences with individual students.

vi) Specific assessment – assessment booklets for each network of activities would provide teachers with opportunities for written feedback.

vii) Notebooks – as they do the activities, have the students record their written responses in a notebook. The amount of written work that they complete as they respond to the activities is often amazing!

Meeting individual needs The concept maps on pages 27-63 provide a valuable vehicle for addressing indi-

vidual needs by showing which concepts need to be understood before later ones can be acquired with understanding. The concept maps can be used diagnostically to provide activities at appropriate levels.

i) Within each activity Individual differences are accommodated by the very nature of the activities. For example, in Crossing (Num 3.2/4, SAIL Volume 1), students roll a die to

move their counters up a 10-square number track. A student whose counter is on the ninth square, just needs to roll a ‘1’ to finish. The other child is at square five. Each child rolls the die; the one at square five gets a ‘5’ and wins the game. The first child says, “How did that happen? I was closer to the end.” The teach-er discusses how probability works when you throw dice. One child may be working on addition: 5 + 5 = 10 whereas the other child is thinking about prob-ability.

ii) Individual differences can also be accommodated by introducing a more ad-vanced activity to a small group while others continue to work on an earlier activity. For example, in Making equal parts (Num 7.1 Activities 1, 2, 4 and 5) one group of students can be making physical representations in Activities 1 and 2 while others move on to pictorial representations in Activity 4 and others can consolidate the concepts in a challenging game, Activity 5.

iii) Because the activities are open ended, students are able to work at their own pace and at a suitable level within each activity. One group of students may complete 20 addition questions using the ‘Start,’ ‘Action,’ and ‘Result,’ cards, while another group, using counters, may only complete 5. Each feels success-ful.

8

Teacher’s guide

Across the curriculum By their very nature, the ways of learning mathematics embodied in this program

both depend upon and contribute to social learning and clear speech.

Many opportunities arise which spontaneously lead to activities across the cur-riculum. For example, in Feeding the animals (Num 7.2/1) students may wish to expand their knowledge about nutrition for zoo animals. Traineekeepers,qualifiedkeepers (Num 7.2/2) could lead to an examination of the different occupations at the zoo. Cargo airships (Num 6.8/4) might be used in conjunction with a study about the airport. Treasure chest (Num 5.10/2) could be used in conjunction with a unit about the sea. The real life situations provided by many of the activities can lead to many and diverse areas of study across the curriculum.

Problem solving Skemp’s theory-based approach to problem solving has students learning one new

concept at a time in the context of activities that are interesting and engaging but low in mathematically irrelevant material. This facilitates the process of abstraction as progress is made through the relevant concept maps and the child’s knowledge struc-ture is developed. The carefully sequenced activities lead to the development of ap-propriate mathematical models which can be used to generate solutions to problems, which in turn are tested in the original problem situation. This approach contrasts strikingly with approaches that begin with high-noise problem situations and lead to a disorderly development of concepts and processes.

Personalized number stories are included in each Network: addition, pages 170-174 in Volume 1; subtraction, pages 212-215 in Volume 1; multiplication, pages 151-154 in Volume 2; and division, pages 260, 268 in Volume 1.

Students learn how to solve problems. They are deliberately led from verbal

problems to physical representations of the objects, numbers, and actions described in the number story (modelling), and from the latter to the mathematical statement, not directly from words to mathematical symbols. They learn how to produce numerical models in physical materials corresponding to given number stories, to manipulate these appropriately, and to interpret the result in the context of the number story. Later, they do this by using written symbols only. (For an example, see Number stories: abstracting number sentences, Num 5.4, Activities 1, 2, 3.)

Students are encouraged to draw diagrams to show what they have done (con-crete-pictorial-symbolic), e.g., Different questions, same answer. Why? (Num 6.3/1), to show the connection between grouping and sharing in division.

As they are engaged in the activities, students use problem solving skills. E.g., in

Treasure chest (Num 5.10/2) they are applying problem solving skills as they pre-dict the best strategies for choosing their treasure.

The students are helped to invent their own real-world problems, applying the concepts they have learned in practical situations. E.g., Front window, rear win-dow – make your own (Num 4.9/4).

9

Teacher’s Guide Meeting the NcTM standards

The close correlation between the NCTM Standards and the SAIL program written by Richard Skemp is undoubtedly the result of the convergence of two independent, thorough, and insightful explorations to the heart of what is needed for the intelligent learning of mathematics.

Accepting that the NCTM Standards have established a broad framework for guiding needed reform in school mathematics, an examination of some of the ways in which the Skemp materials fit that framework follows.

Mathematics as Problem Solving (Standard 1). The Skemp learning activities are, themselves, problem-solving tasks. The students are led to construct mathemati-cal concepts and relationships from physical experiences designed to appeal to their imagination and to build on their real-world experiences. Students work coopera-tively on well-designed, goal-directed tasks, making predictions, testing hypotheses and building relational understandings that facilitate routine and non-routine problem solving. (This is addressed at length above.)

Mathematics as Communication (Standard 2). The Skemp learning activities are designed to foster communication about mathematical concepts between students and between students and adults. A typical cooperative-group activity, Number targets (Num 2.8/1), engages the students in communicating about place value con-cepts with physical embodiments, spoken symbols, written symbols, oral and read symbols, all the while exploring the underlying mathematical meanings while using problem solving strategies to predict their best move.

Mathematics as Reasoning (Standard 3). Using patterns and relationships to make sense out of situations is an integral component of the Skemp learning activities. As mentioned previously, the activities frequently lead the students to explore, conjec-ture (make predictions), and test their conjectures. The program builds on relational understanding, and useful instrumental (habit) learning is promoted when appropri-ate.

Mathematical Connections (Standard 4). Skemp’s detailed conceptual analysis of the elementary school mathematics curriculum has produced a set of well-defined concept maps or networks. The networks arrange the activities in optimal learning sequences and provide teachers with the framework to make relational connections within and across networks. Many of the activities require assembling previously learned concepts and processes to deal with the task at hand. An entire network (NuSp 1, The number track and the number line) is devoted to number tracks and number lines, which are of importance throughout mathematics, from kindergarten through university-level mathematics, and beyond. They provide a valuable support for our thinking about numbers in the form of a pictorial representation. Skemp’s unerring notions about contexts that appeal to student imaginations have produced interesting lifelike settings in which the students learn. They compare possible out-comes of the moves they might make in One tonne van drivers (Num 3.10/3), and they are introduced to a budgeting activity in Catalogue shopping (Num 3.10/4).

2

10

Teacher’s guide

In adult life, planning the use of money and other resources – e.g., time, labour – is one of the major uses of arithmetic. Because of interesting real life situations, connections to other curriculum areas are easily integrated. Feeding the animals (Num 7.2/1) and Setting the table (Num 4.5/2, SAIL Volume 1) are examples of activities which have spin-offs to art and health.

Estimation, Number Sense & Numeration, Numbers & Operations, Computation, Number Systems (Standards 5-7). One entire network of Skemp learning activi-ties, Num 1, treats Numbers and their properties from ‘sorting dots’ to ‘square numbers’ and ‘relations between numbers.’ Another complete network, Num 2, The naming of numbers, carefully builds everything one needs to know about place val-ue for numerals of any number of digits. A network is devoted to each of the basic arithmetic operations, Addition, Subtraction, Multiplication, and Division. In all of these networks there is emphasis on number sense, operations sense, reasonable estimates, making and testing predictions, mental computation, thinking strategies, and relationships between concepts and between operations . . . all recommended in Standards 5 through 8. Calculators are used when appropriate, as they would be in real-life situations. Activities which use calculators include:

One tonne van drivers (Num 3.10/3), Cargo boats (Num 5.7/3), Treasure chest (Num 5.10/2), Cargo airships (Num 6.8/4), Number targets: division by calcula-tor (Num 6.9/1), Predict, then press (Num 7.8/2), “Are calculators clever?” (Num 7.8/3), and Number targets by calculator (Num 7.8/4).

Geometry, Measurement (Standards 9, 10 [Grades K-4]; 12, 13 [Grades 5-8]). Space 1 and the new Space 2 network cover properties and components of two- and three-dimensional shapes, symmetry, and motion geometry. NuSp 1, The number track and the number line network, develops basic linear measurement skills while providing useful embodiments for the activities developing number concepts and arithmetic operations. The SAIL measurement networks (Meas 1, 2, 3, 4, 5, and 6) did not appear in the Structured Activities for Primary Mathematics volumes published in England by Routledge in 1989. Their treatment of Length, Area, Vol-ume & capacity, Mass & weight, Time, and Temperature covers the concepts and relationships of Standards 9 & 10 for Grades K-4 and Standards 12 & 13 for Grades 5-8.

Statistics, Probability (Standards 11 [Grades K-4]; 10, 11 [Grades 5-8]). The prereq-uisite skills have been well developed in the ten original networks where students are introduced to probability from real-life situations. For example, as previously dis-cussed (see page 7), in Crossing (Num 3.2/4, SAIL Volume 1), students roll a die to move their counters up a 10-square number track. A student whose counter is on the ninth square, just needs to roll a ‘1’ to finish. The other child is at square five. Each child rolls the die; the one at square five gets a ‘5’ and wins the game. The first child says, “How did that happen? I was closer to the end.” The teacher discusses how probability works when you throw a die. One child may be working on addition: 5 + 5 = 10 whereas the other child is thinking about probability.

11

Teacher’s Guide Fractions and Decimals (Standard 12 [Grades K-4]; Standards 5, 6 [Grades 5-8]).

The Num 7 network, Fractions, begins with the development of real-object con-cepts of ‘equal parts,’ ‘denominators,’ and ‘numerators,’ building to fully-symbolic treatments of fractions, decimal-fractions, and operations with decimals. The whole development of fraction ‘number sense’ is firmly grounded in the use of concrete and pictorial models. The relationships between fractions and decimal fractions are care-fully developed.

The standards and Professional development

Inservice courses, based on Skemp’s learning theory, have been developed to give teachers opportunities to learn about the theory, to do a selection of activities togeth-er, to make them, and to discuss them. Follow-up professional development support is available as teachers incorporate the activities in their classrooms. As mentioned previously, thirty-nine VHS videoclips of the SAIL activities and of the theory which they embody have been produced as part of that support. The SAIL program is well suited for addressing the Standards in a thorough, consistent, and well-organized manner.

Each of these activities embodies both a mathematical concept, and also one or more aspects of the theory. So by doing these with a group of children, both children and their teacher benefit. The children benefit by this approach to their learning of mathematics; and the teacher also has an opportunity to learn about the theory of intelligent learning by seeing it in action. Theoretical knowledge acquired in this way relates closely to classroom experience and to the needs of the classroom. It brings with it a bonus, since not only do the children benefit from this approach to math-ematics, but it provides a good learning situation for teacher also. In this way we get ‘two for the price of one’, time-wise.3

One ship sails East, another West by the self-same winds that blow. It isn’t the gales, but the trim of their sails that determines the way they go. Traditional

Notes

1 Skemp, R. R. (1989). Mathematics in the Primary School. London: Routledge. p. 166.

2 National Council of Teachers of Mathematics. (1989). Curriculum and Evalua-tion Standards for School Mathematics. Reston, Va.: The Council.

3 Skemp, 1989, op. cit., p. 111.

12

Teacher’s guide

13

Introduction

IntroductIon

1 Why this book was needed and what it provides

The Curriculum and Evaluation Standards for School Mathematics1 document pro-duced by the National Council of Teachers of Mathematics stresses the importance of a developmentally appropriate curriculum. There is now a wide consensus that practical work is essential for ensuring developmentally appropriate concept for-mation throughout the elementary school years, and not just for younger children. There are now a number of these activities available, and individually many of them are attractive and worthwhile. But collectively, they lack two essential requirements for long-term learning: structure, and clear stages of progression. SAIL through Mathematics, Volumes 1 and 2, provides a fully structured collection of more than four hundred seventy-five activities, covering a core curriculum for children aged from four to eleven years old and using practical work extensively at all stages.

This collection is not, however, confined to practical work. Mathematics is an abstract subject, and children will need in the future to be competent at written math-ematics. Putting one’s thoughts on paper can be a help in organizing them, as well as recording them for oneself and communicating them to others. What is important is that this should not come prematurely. It is their having had to memorize a collec-tion of rules without understanding which has put so many generations of learners off mathematics for life, and destroyed their confidence in their ability to learn it. Practical, oral, and mental work can provide the foundation of understanding with-out which written work makes no sense. Starting with these, the present collection provides a careful transition from practical work to abstract thinking, and from oral to written work.

Activities for introducing new concepts often include teacher-led discussion. Many of the other activities take the form of games which children can play together without direct supervision, once they know how to play. These games give rise to discussion; and since the rules and strategies of the games are largely mathematical, this is a mathematical discussion. Children question each other’s moves, and justify their own, thereby articulating and consolidating their own understanding. Often they explain things to each other, and when teaching I emphasize that “When we are learning it is good to help each other.” Most of us have found that trying to explain something to someone else is one of the best ways to improve one’s own understand-ing, and this works equally well for children.

This volume also provides the following: (a) A set of diagrams (concept maps, or networks) showing the overall mathematical

structure, and how each topic and activity fits. (b) Clear statements of what is to be learned from each group of activities. (c) For each activity, a list of materials and step by step instructions. (In Volume 2a,

photomasters are also provided to simplify the preparation of materials.) (d) For each topic, discussion of the mathematical concept(s) involved, and of the

learning processes used. The last of these will, it is hoped, be useful not only for classroom teachers, but also

for support teams, mathematics advisers, those involved in the pre-service and in-service education of teachers, and possibly also those whose main interests are at the research level.

14

Introduction 2 the invisible components and how to perceive them

The activities in this collection contain a number of important components which are invisible, and can only be perceived by those who know what to look for. These include (i) real mathematics, (ii) structure, and (iii) a powerful theory.

(i) Real mathematics. I contend that children can and should enjoy learning real mathematics. You might ask: “What do you mean by this? Is it just a puff?” I say “Begin,” because a fuller answer depends on personal experience. If someone asks “What is a kumquat?” I can tell them that it is a small citrus fruit, but two of the most important things for them to know are what it tastes like, and whether they like it or not. This knowledge they can only acquire by personally tasting a kumquat.

Real mathematics is a kind of knowledge. I can describe it, and I hope you will find this a useful start. But some of the most important things about mathematics people cannot know until they have some of this kind of knowledge in their own minds; and those who acquired real mathematics when they were at school are, regrettably, in a minority. A simple preliminary test is whether you enjoy mathemat-ics, and feel that you understand it. If the answers are “No,” then I have good news for you: what you learned was probably not real mathematics. More good news: you can acquire real mathematics yourself while using these activities with your children. You will then begin to perceive it in the activities themselves: more accurately, in your own thinking, and that of your children, while doing these activities. And you will begin to discover whether or not what you yourself learned as a child was real mathematics.

Mathematics (hereafter I will use ‘mathematics’ by itself to mean real mathemat-ics) is a kind of knowledge which is highly adaptable. In the adult world, this adapt-ability can be seen in the great variety of uses to which it is put. Mathematics is used to make predictions about physical events, and greatly increases our ability to achieve our goals. Our daily comfort and convenience, sometimes our lives, depend on the predictive use of mathematics by engineers, scientists, technicians, doctors and nurses. At an everyday level, we use mathematics for purposes such as predict-ing approximately how long we should allow for a journey. Highly sophisticated mathematics is required to project communication satellites into orbits whereby they hang stationary relative to the earth; and also in the design of the satellites them-selves, whose electronic equipment allows us to watch on our television screens events many thousands of miles away.

Mathematics has also an important social function, since many of the complex ways in which we co-operate in modern society would not be possible without mathematics. Nuts could not be made to fit bolts, clothes to fit persons, without the measurement function of mathematics. Businesses could not function without the mathematics of accountancy. If the person in charge of this gets his calculations wrong, his firm may go out of business: that is to say, others will no longer cooperate by trading with them.

Another feature of mathematics is creativity — the use of one’s existing knowl-edge to create new knowledge. Can you say what are ninety-nine sevens? Probably not, but if you think “A hundred sevens make seven hundred, so ninety-nine sevens will be one seven less: six hundred and ninety three,” then you are using your own

15

Introductionmental creativity. Creating new mathematics which nobody ever knew before is cre-ativity at the level of the professional mathematician; but anyone who has some real mathematics is capable of creating knowledge which is new to them, and this way of using one’s mind can give a kind of pleasure which those who have not experienced it may find hard to understand.

These are some of the adult uses of mathematics, which make it so important in today’s world of advanced science, technology, and international commerce. At school, most children still learn a look-alike which is called by the same name, but whose uses have little in common with the uses of real mathematics. School math-ematics as it is experienced by children is mostly for getting check marks, pleasing teachers, avoiding reproofs and sometimes also the humiliation of being made to feel stupid. It is also used for passing exams, and thereafter quickly forgotten. Yet real mathematics can be taught and learned at school. For an example of mathematics used predictively, try Missing stairs (Org 1.5/1, SAIL Volume 1). Success in most of the games also depends largely on making good predictions. Mathematics is used socially in all children’s work together in groups; and in some, e.g., Renovating a house (Num 3.9/3), a social use is embodied in the activity itself. I hope that you will find pleasure in discovering examples of creativity in the thinking of your own children when they are learning real mathematics in contexts like these.

(ii) Structure. This is an essential feature of real mathematics. It is this which

makes possible all the features described in (i), so for emphasis I am giving it a sec-tion to itself.

By structure we mean the way in which parts fit together to make a whole. Often this whole has qualities which go far beyond the sum of the separate properties of the parts. Connect together a collection of transistors, condensers, resistors, and the like, most of which will do very little on their own, and you have a radio by which you can hear sounds broadcast from hundreds of miles away. That is, if the connections are right: and this is what we mean by structure in the present example.

In the case of mathematics, the components are mathematical concepts, and the structure is a mental structure. This makes it much harder to know whether it is there or not in a learner’s mind. But the difference between a mathematical structure and a collection of isolated facts is as great as the difference between a radio and a box of bits. There is the same difference between a radio set and a wrongly connect-ed assortment of components, but this is harder to tell by looking. The important test of the presence or absence of structure, i.e. of the right set of connections, may best be inferred from performance. This is also true for mathematics, and for its look-alike which goes by the same name. Of these two, only real mathematics performs powerfully, enjoyably, and in a wide variety of ways.

Each individual learner has to put together these structures in his own mind. No one can do it for him. But this mental activity can be greatly helped by good teach-ing, an important part of which is providing good learning situations.

The requirements of a good learning situation include full use of all of the three modes of building conceptual structures. Mode 1 is learning by the use of practi-cal materials; Mode 2 is learning from exposition, and by discussion; and Mode 3 is expanding one’s knowledge by creative thinking. These categories are expanded and discussed more fully in Mathematics in the Primary School.2

16

Introduction The activities in this book are intended to help teachers provide learning situations

of the kind described. They are also fully structured, meaning that the concepts embodied in each fit together in ways which help learners to build good mathemati-cal structures in their own minds. This also includes consolidation, and developing mathematical skills.

(iii) A powerful theory. In 1929, Dewey wrote “Theory is in the end . . . the most

practical of all things”;3 and I have been saying the same for many years, even before I knew that Dewey had said it first. The activities in this book embody a new theory of intelligent learning. This had its origins in the present author’s researches into the psychology of learning of mathematics,4 and was subsequently expanded and generalized into a theory of intelligent learning which can be applied to the learning of all subjects.5 It is not essential to know the theory in order to use the activities. But readers who are professional teachers will want to know not only what to do but why. Mathematics advisers, or lecturers in mathematical education, will wish to satisfy themselves of the soundness of the underlying theory before recommending the activities.

This theoretical understanding is best acquired by a combination of first-hand experi-ence, reading, and discussion. Each of the activities embodies some aspects of the general theory, so by doing the activities with children we can observe the theory in action. For school teachers, this is a very good way to begin, since the theoretical knowledge acquired in this way begins with classroom experience, and as it develops further will continue to relate to it. This also has the advantage that we get ‘two for the price of one,’ time-wise: what might be called a ‘happy hour’ in the classroom! These observations can then form the first part of the trio

OBSERVE AND LISTEN REFLECT DISCUSS

whose value for school-based inservice education will be mentioned again in Section 5.

Reading helps to organize our personal experience, and to extend our knowledge beyond what can be gathered first-hand. A companion book to the present volume is Mathematics in the Primary School,6 and this also offers suggestions for further reading.

17

Introduction 3 How this book is organized and how to use it

The two SAIL volumes contain teaching materials for eight school years, together with explanations and discussions. This is a lot of information. Careful thought has therefore been given to its organization, to make it easy to find as much as is required at a given stage, and to avoid feeling overloaded with information. Math-ematics is a highly concentrated kind of information, so it is wise to take one’s time, and to go at a pace which allows comfortable time for assimilation. The amount eventually to be acquired in detail by a class teacher would be no more than one-eighth of the total, if all children in the class were of the same ability. In practice it will, of course, be more because of children’s spread of ability.

The aim has been to provide first an overview; then a little more detail; and then a lot of detail, of which many readers will not need all, nor all at the same time. This has been done by organizing the subject matter at four levels, into THEMES, NETWORKS, TOPICS, and ACTIVITIES. The themes and networks are tabulated below.

THEMES NETWORKS CODES

Organizing Set-based organization Org 1

Number Numbers and their properties Num 1 The naming and recording of numbers Num 2 Addition Num 3 Subtraction Num 4 Multiplication Num 5 Division Num 6 Fractions Num 7

Space Shape Space 1 Symmetry and motion geometry Space 2

Synthesis of The number track and the number line NuSp 1 Number and Space

Pattern Patterns Patt 1 Measurement Length Meas 1 Area Meas 2

Volume and capacity Meas 3 Weight and mass Meas 4 Temperature Meas 5 Time Meas 6

18

Introduction The five main themes run in parallel, not sequentially, though some will be started

later than others. Within the theme of Number, there are seven networks. For some themes, there is at present just one network each; but I have kept to the same ar-rangement for consistency, and to allow for possible future expansion. By ‘network,’ I mean a structure of interrelated mathematical ideas. It can well be argued that all mathematical ideas are interrelated in some way or other, but the networks help to prevent information overload by letting us concentrate on one area at a time.

Greater detail for each network is provided by a concept map and a list of activi-ties. These are on pages 27 through 63. It will be useful to look at a pair of these as illustrations for what follows. Each concept map shows how the ideas of that net-work relate to each other, and in particular, which ones need to be understood before later ones can be acquired with understanding. These interdependencies are shown by arrows. A suggested sequence through the network is shown by the numbers against each topic. Use of the concept map will help you to decide whether other sequences may successfully be followed, e.g., to take advantage of children’s current interests. The concept map is also useful diagnostically. Often a difficulty at a par-ticular stage may be traced to a child’s not having properly understood one or more earlier concepts, in which case the concept map will help you to find out which these are. Another function of the concept maps is to help individual teachers to see where their own teaching fits into a long-term plan for children’s learning, throughout the primary years: what they are building on, and where it is leading.

Below each concept map is shown a list of the activities for each topic. (I call them ‘topics’ rather than ‘concepts’ because some topics do not introduce new con-cepts, but extend existing ones to larger areas of application.) Usually these activi-ties should be used in the order shown. An alphabetical list of the activities is also given, at the end of the book.

To find activities for a particular topic, the best way is via the concept maps and the lists of activities opposite them. Suppose for example that you want activities for adding past ten. For this you naturally look at the concept map for addition, and find adding past ten as topic 7. On the adjacent page, for this topic, you will find seven activities. Not all topics have so many activities, but this indicates the importance of this stage in children’s learning of addition. If on the other hand you want to find a particular activity by name, then the alphabetical index at the end of the book will enable you to do so.

The codes for each topic and activity are for convenience of reference. They show where each fits into the whole. Thus Num 3.8/2 refers to Network Num 3, Topic 8, Activity 2. If the packet for each activity has its code on it, this will help to keep them all in the right order, and to replace each in the right place after use.

19

Introduction 4 Getting started as a school

Since schools vary greatly, what follows in this section and the rest of this introduc-tion is offered as no more than a collection of suggestions based on the experience of a number of the schools where the materials have already been introduced.

It has been found useful to proceed in two main stages: getting acquainted, and full implementation. Since the latter will be spread over one or two years at least, the first stage is important for getting the feel of the new approach, and to help in deciding that it will be worth the effort.

For getting acquainted, a good way is for each teacher to choose an activity, make it up, and learn it by doing it with one or more other teachers. (Different activities are for different numbers of persons.) Teachers then use these activities with their own children, and afterwards they discuss together what they have learned from observation of their own children doing the activities. It is well worth while trying to see some of the activities in use, if this can be arranged. Initially, this will convey the new approach more easily than the printed page.

When you are ready to move towards a full implementation, it will be necessary to decide the overall approach. One way is to introduce the activities fully into the first and second years, while other teachers gradually introduce them into later years as support for the work they are already doing. This has the advantage that the full implementation gradually moves up the school, children being used to this way of learning from the beginning. Alternatively, activities may be introduced gradually throughout the school, individual teachers choosing which activities they use along-side existing text-based materials while they gain confidence in the new approach. As another alternative, a school may wish to begin with one network (perhaps, for example, multiplication), arranging for all the teachers to meet together to do all of the activities in the network and then to implement them at the appropriate grade levels.

Arrangements for preparation of the materials need to be planned well in advance. This is discussed in greater detail in section 5. A detail which needs to be checked in good time is whether the commercial materials needed, such as Multilink or Unifix and base ten materials, are already in the school in the quantities needed.

In considering the approach to be used, it is important to realize that while benefit is likely to be gained from even a limited use of the activities, a major part of their value is in the underlying structure. The full benefit, which is considerable, will therefore only be gained from a full implementation.

20

Introduction 5 organization within the school

Overall, the organization of the new approach is very much a matter for the principal and staff of each individual school to work out for themselves; so, as has already been emphasized, what follows is offered only in the form of suggestions, based on what has been found successful in schools where this approach has been introduced.

Whatever organization is adopted, it is desirable to designate an organizer who will coordinate individual efforts, and keep things going. It is a great help if this teacher can have some free time for planning, organizing, advising, and supporting teachers as need arises.

One approach which has been found successful is as follows. The mathematics or-ganizer holds regular meetings with the staff in each one- or two-year group, accord-ing to their number. Each teacher chooses an activity, makes it up if necessary, and teaches it to the other teachers in the group. They discuss the mathematics involved, and any difficulties. Subsequently they discuss their observations of their own chil-dren doing these activities, and what they have learned by reflecting on these. This combination may be summarized as

OBSERVE AND LISTEN REFLECT DISCUSS

and is an important contribution to school-based inservice education. (So much so, that several leading mathematics educators have said that it should be printed on every page. As a compromise, I have printed it at the end of every topic.)

It is good if the mathematics organizer is also in a position to help individual teachers, since it is only to be expected for them to sometimes feel insecure when teaching in a style which may be very different from that to which they are accus-tomed. Two useful ways to help are by looking after the rest of a teacher’s class for a while, so that this teacher is free to concentrate entirely on working with a small group; and by demonstrating an activity with a small group while the class teacher observes, the rest of the class being otherwise occupied.

21

Introduction 6 Getting started as an individual teacher

The most important thing is actually to do one or more activities with one’s own children, as early on as possible. This is the best way to get the feel of what the new approach is about. After that, one has a much better idea of where one is going. If there are particular topics where the children need help, suitable activities may be found via the concept maps and their corresponding lists of activities. Alternatively, here is a list of activities which have been found useful as ‘starters’. The stages cor-respond roughly to years at school.

vol page Stage 1 Lucky dip Org 1.3/1 1 72 “Can I fool you?” Org 1.3/2 1 73 Missing stairs Org 1.5/1 1 76 Stage 2 Stepping stones Num 3.2/3 1 162 Crossing Num 3.2/4 1 164 Sequences on the number track NuSp 1.2/1 1 309 Stage 3 The handkerchief game Num 3.5/1 1 175 “Please may I have?” (complements) Num 3.5/2 1 176 Number targets Num 2.8/1 1 141 Stage 4 Slippery slope Num 3.7/3 1 184 Slow bicycle race NuSp 1.5/1 1 319 Doubles and halves rummy Num 1.10/3 1 121 Stage 5 Place value bingo Num 2.10/3 2 91 Renovating a house Num 3.9/3 2 117 Constructing rectangular numbers Num 6.4/1 2 194 The rectangular numbers game Num 6.4/2 2 195 Stage 6 Cycle camping Num 3.10/2 2 121 One tonne van drivers Num 3.10/3 2 122 Multiples rummy Num 5.6/6 2 168 Stage 7 Cargo boats Num 5.7/3 2 172 Classifying polygons Space 1.12/1 2 278 Match and mix: polygons Space 1.12/3 2 279

When the children are doing an activity, think about the amount of mathematics which the children are doing, including the mental and oral activity as well as the written work, and compare it with the amount of mathematics they would do in the same time if they were doing written work out of a textbook.

So far, I have interpreted the heading of this section as meaning that the reader is an individual teacher within a school where most, or at least some, colleagues are also introducing the new approach. But what if this is not the case? When talking with teachers at conferences, I have met some who are the only ones in their school who are using this kind of approach to the teaching of mathematics.

This is a much more difficult situation to be in. We all need support and encour-agement, especially when we are leaving behind methods with which we are familiar — even though they have not worked well for many children. We need to discuss difficulties, and to share ways we have found for overcoming them. So my sugges-tion here is that you try to find some colleague with whom you can do this. At the very least, you need someone with whom to do the activities before introducing them to the children; and further discussions may arise from this.

22

Introduction 7 Parents Parents are naturally interested in their children’s progress at school. Written work

is something they can see — what they cannot see is the lack of understanding which so often underlies children’s performances of these ‘rules without reasons.’ Some-times also they try to help children at home with their mathematics. Unfortunately, this often takes the form of drill-and-practice at multiplication tables, and pages of mechanical arithmetic. This is the way they were taught themselves, and some par-ents have been known to respond unfavourably when their children come home and say that they have spent their mathematics lessons playing games. As one teacher reported, “Games are for wet Friday afternoons. Mathematics is hard work. They aren’t meant to enjoy it.”

How you deal with problems of this kind will, of course, depend partly on the nature of existing parent-teacher relationships in your own school. When explaining to parents who may be critical of what you are doing, it also helps if you are confi-dent in your own professional understanding, and if there is a good consensus within the school. These are areas in which sections 2 and 5 offer suggestions. Some ap-proaches which have been used with success are described here. They may be used separately or together.

A parents’ evening may be arranged, in which parents play some of the games together. Teachers help to bring out the amount of mathematics which children are using in order to decide what move they will make, or what card to play. To do this well, teachers need to be confident in their own knowledge of the underlying math-ematics, and this can be built up by doing the activities together, and discussing with each other the mathematics involved.

A small group of parents may be invited to come regularly and help to make up activities. This needs careful organizing initially, but over a period it can be a great saving in teacher’s time. When they have made up some activities, parents naturally want to do them in order to find out what they are for; so this combines well with the first approach described.

Some parents may also be invited to come into classrooms and supervise consoli-dation activities. (It has already been mentioned that activities which introduce new concepts should be supervised by a teacher.)

For parents who wish to help children at home with their mathematics, the games provide an ideal way to do this. Many of these can be played at varying levels of so-phistication, which makes them suitable as family games; and there is usually also an element of chance, which means that it is not always the cleverest player who wins. None of the games depends entirely on chance, however. Good play consists in making the best of one’s opportunities. Parents who help their children in this way will also have the benefit of knowing that what they are doing fits in with the ways in which their children are learning at school.

23

Introduction 8 Some questions and answers

Q. How long will it take to introduce these materials? A. The materials embody many ideas which are likely to be new to many teachers,

in two areas: mathematics, and children’s conceptual learning. So it is important to go at a pace with which one feels comfortable, and which gives time to assimilate these ideas into one’s personal thinking and teaching. It took me twenty-five years to develop the underlying theory, three to find out how to embody it in ways of teaching primary mathematics, and another five to devise and test the integrated set of curriculum materials in this volume. If a school can have the scheme fully imple-mented and running well in about two years, I would regard this as good going. But of course, you don’t have to wait that long to enjoy some of the rewards.

Q. Isn’t it a lot of work? A. Changing over to this new approach does require quite a lot of work, especially

in the preparation of the materials. The initial and ongoing planning and organiza-tion are important, so that this work can go forward smoothly. As I see it, nearly all worthwhile enterprises, including teaching well, involve a lot of work. If you are making progress, this work is experienced as satisfying and worth the effort. As one teacher said, ‘Once you get started, it creates its own momentum.’ And once it is established, the fact that children are learning more efficiently makes teaching easier.

Q. How were the concept maps constructed? A. First, by a careful analysis of the concepts themselves, and how they relate to

each other in the accepted body of mathematical knowledge.7 This was then tested by using it as a basis for teaching. If an activity was unsuccessful in helping children to develop their understanding, this was discussed in detail at the next meeting with teachers. Sometimes we decided that the activity needed modification; but some-times we decided that the children did not have certain other concepts which they needed for understanding the one we had been introducing – that is, the concept map itself needed revision. So the process of construction was a combination of math-ematics, applied learning theory, and teaching experiment.

Q. Is it all right to use the activities in a different order? A. Within a topic, the first activity is usually for introducing a new concept, and

clearly this should stay first. Those which follow are sometimes for consolidation, in which case the order may not matter too much. Sometimes, however, they are for developing thinking at a more abstract level, in which case the order does matter. However, once the activities have been used in the order suggested, it is often good to return to earlier ones, for further consolidation and to develop connections in both directions: from concrete to abstract, and from abstract to concrete.

Whether it is wise to teach topics in a different order you can decide by looking at the concepts map itself. These, and the teaching experiments described, show that the order of topics is very important. They also show that for building up a given knowledge structure, there are several orders which are likely to be successful – and many which are likely to be unsuccessful.

24

Introduction Q. Is it all right to modify the activities? A. Yes, when you are confident that you understand the purpose of the activity and

where it fits into the long-term learning plan. The details of every activity have been tested and often rewritten several times, both from the point of view of the math-ematics and to help them to go smoothly in the classroom; so I recommend that you begin by using them as written. When you have a good understanding of the math-ematics underlying an activity, you will be in a position to use your own creativity to develop it further.

Sometimes children suggest their own variation of a game. My usual answer is that they should discuss this among themselves, and if they agree, try it together next time: but that rules should not be changed in the middle of a game. They may then discuss the advantages and disadvantages of the variation.

Q. Can I use the materials alongside an existing scheme? And what about written work, in general?

A. Especially with the older children, I would expect you to begin by introducing these activities alongside the scheme you are already using and familiar with. With younger children, the activities introduce as much written work as I think is neces-sary. Thereafter, existing textbook schemes can be put to good use for gradually in-troducing more written work of the conventional kind. But these should come after the activities, rather than before. Children will then get much more benefit from the written work because they come to it with greater understanding.

Q. Doesn’t all this sound too good to be true? A. Frankly, yes. Only personal experience will enable you to decide whether it can

become true for you yourself, and in your school. Each school has its own micro-climate, within which some kinds of learning can thrive and others not. Where the microclimate is favourable to this kind of learning, what I have been describing can become true, and I think you will find it professionally very rewarding. Where this is not at present the case, the problems lie beyond what can be discussed here. I have discussed them at length elsewhere.8

Q. What do you see as the most important points when implementing this ap-proach?

A. Good organization; personal experience of using the activities; observation, reflection and discussion.

25

Introduction notes

1 National Council of Teachers of Mathematics. (1989). Curriculum and Evalua-tion Standards for School Mathematics. Reston, Va.: The Council.

2 Skemp, R. R. (1989). Mathematics in the Primary School. London: Routledge. 3 Dewey, J. (1929). Sources of a Science of Education. New York: Liveright. 4 Skemp, R. R. (1971, 1986 2nd ed.). The Psychology of Learning Mathematics.

Harmondsworth: Penguin. 5 Skemp, R. R. (1979). Intelligence, Learning, and Action. Chichester: Wiley. 6 Skemp, 1989, op. cit. 7 Skemp, 1971 and 1978, op. cit. See Chapter 2 of the first edition, or Chapter 1 of

the second edition of Skemp, 1971. 8 Skemp, 1979, op. cit., Chapter 15.

26

Introduction

27

ConCept Maps and Lists of aCtivities

Activities in bold are those found in this volume.

28

Org 1 Set-based organization

29

1 Sorting 651/1 Perceptual matching of objects 651/2 Matching pictures 651/3 A picture matching game 661/4 Dominoes 661/5 Conceptual matching 671/6 Conceptual matching 671/7 Attribute cards 68

2 Sets 692/1 Introduction to Multilink or Unifix 692/2 Making picture sets 692/3 “Which set am I making?” 702/4 “Which two sets am I making?” 70

3 Comparing sets by their numbers 723/1 Lucky dip 723/2 “Can I fool you?” 73

4 Ordering sets by their numbers 744/1 Ordering several rods by their lengths 744/2 Combining order of number, length, and position 74 5 Complete sequences of sets 765/1 Missing stairs 76

6 The empty set; the number zero 786/1 The empty set 78

7 Pairing between sets 807/1 Physical pairing 807/2 Mentally pairing 81

8 Sets which match, counting, and number 828/1 Sets which match 83

9 Counting, matching, and transitivity 84 No activity; but a note for teachers

10 Grouping in threes, fours, fives 8610/1 Making sets in groups and ones 8710/2 Comparing larger sets 8810/3 Conservation of number 88

11 Bases: units, rods, squares and cubes 9011/1 Units, rods and squares 9111/2 On to cubes 92

12 Equivalent groupings: canonical form 9312/1 “Can I fool you?” (Canonical form) 9412/2 Exchanging small coins for larger 95

13 Base ten 6713/1 Tens and hundreds of cubes 6713/2 Tens and hundreds of milk straws 6813/3 Thousands 68

Org 1 Set-based organization

30 Num 1 Numbers and their properties

31

1 Sets and their numbers perceptually (subitizing) 991/1 Sorting dot sets and picture sets 991/2 Picture matching game using dot sets and picture sets 100

2 Successor: notion of one more 1012/1 Making successive sets 1012/2 Putting one more 101

3 Complete numbers in order 1033/1 “Which card is missing?” 103

4 Counting 1044/1 Finger counting to 5 1044/2 Planting potatoes 105

5 Extrapolation of number concepts to 10 1065/1 Finger counting to 10 1065/2 Missing stairs, 1 to 10 1065/3 “I predict-here” on the number track (Same as NuSp 1.1/1) 1075/4 Sequences on the number track (Same as NuSp 1.2/1) 109

6 Zero 1106/1 “Which card is missing?” (Including zero) 1106/2 Finger counting from 5 to zero 110

7 Extrapolation of number concepts to 20 1127/1 Finger counting to 20: “Ten in my head” 112 8 Ordinal numbers first to tenth 1138/1 “There are . . . animals coming along the track” 1138/2 “I’m thinking of a word with this number of letters.” 1148/3 “I think that your word is . . .” 115

9 Odds and evens 1169/1 “Yes or no?” 1169/2 “Can they all find partners?” 1179/3 “Odd or even?” 117

10 Doubling and halving 11910/1 “Double this and what will we get?” 11910/2 “Break into halves, and what will we get?” 12010/3 Doubles and halves rummy 121

11 Extrapolation of number concepts to 100 6911/1 Throwing for a target 6911/2 Putting and taking 71

12 Ordinal numbers, first to one hundredth 7212/1 Continuing the pattern 7212/2 Sorting proverbs 72 13 Rectangular numbers 7413/1 Constructing rectangular numbers [Same as Num 6.4/1] 7413/2 The rectangular numbers game [Same as Num 6.4/2] 75 14 Primes 7614/1 Alias prime [Same as Num 6.5/3] 7614/2 The sieve of Eratosthenes [Same as Num 6.5/4] 7714/3 Sum of two primes 77

15 Square numbers 7915/1 Square numbers 7915/2 An odd property of square numbers 79

16 Relations between numbers 8216/1 “Tell us something new.” 8216/2 “How are these related?” 83

Num 1 Numbers and their properties

32 Num 2 The naming of numbers

14rounding

33

1 The number words in order (spoken) 1261/1 Number rhymes 126

2 Number words from one to ten 1282/1 Number rhymes to ten 128

3 Single-digit numerals recognized and read 1293/1 Saying and pointing 1293/2 “Please may I have . . .?” 1293/3 Joining dots in order, to make pictures 1303/4 Sets with their numbers 1303/5 Sequencing numerals 1 to 10 131

4 Continuation of counting: 1 to 20 1324/1 Number rhymes to twenty 132

5 Counting backwards from 20 1335/1 Backward number rhymes 1335/2 Numbers backwards 134

6 Counting in twos, fives 1366/1 Counting with hand clapping 1366/2 Counting 2-rods and 5-rods 1366/3 Counting money, nickels 1376/4 Counting sets in twos and fives 137

7 Extrapolation of counting pattern to one hundred 1387/1 Counting in tens 1387/2 Counting two ways on a number square 1387/3 Tens and ones chart 139

Num 2 The naming of numbers

8 Written numerals 20 to 99 using headed columns 858/1 Number targets 86 8/2 Number targets beyond 100 87

9 Written numerals from 11 to 20 889/1 Seeing, speaking, writing 11-19 889/2 Number targets in the teens 89

10 Place-value notation 9010/1 “We don’t need headings any more.” 9010/2 Number targets using place-value notation 9110/3 Place-value bingo 91

11 Canonical form 9411/1 Cashier giving fewest coins 9511/2 “How would you like it?” 97

12 The effects of zero 9912/1 “Same number, or different?” 9912/2 Less than, greater than 100

13 Numerals beyond 100, written and spoken 10213/1 Big numbers 10213/2 Naming big numbers 104

14 Rounding (whole numbers) 10714/1 Run for shelter 10714/2 Rounding to the nearest hundred or thousand 10914/3 Rounding big numbers 111

34 Num 3 Addition

35

1 Actions on sets: putting more (Total <10) 1581/1 Start, Action, Result (do and say) 1581/2 Putting more on the number track (verbal) [NuSp 1.3/1] 160

2 Addition as a mathematical operation 1612/1 Predicting the result (addition) 1612/2 “Where will it come?” (Same as NuSp 1.3/2) 1622/3 Stepping stones 1622/4 Crossing (Same as NuSp 1.3/3) 164

3 Notation for addition: number sentences 1673/1 Writing number sentences for addition 1673/2 Write your prediction (addition) 168

4 Number stories: abstracting number sentences 1704/1 Personalized number stories 1704/2 Abstracting number sentences 1714/3 Personalized number stories - predictive 172 5 Complementary numbers 1755/1 The handkerchief game 1755/2 “Please may I have?” (complements) 176

6 Missing addend 1776/1 “How many more must you put?” 1776/2 Secret adder 1786/3 Personalized number stories: what happened? 179

7 Adding past 10 1817/1 Start, Action, Result over ten 1817/2 Adding past 10 on the number track (Same as NuSp 1.3/4) 1837/3 Slippery slope 1847/4 Addfacts practice 1857/5 Addfacts at speed 1857/6 Predictive number sentences past 10 1877/7 Explorers 187

8 Commutativity 1908/1 Introducing commutativity 1918/2 Introducing non-commutativity 1928/3 Using commutativity for counting on 1928/4 Commutativity means less to remember 193

9 Adding, results up to 99 1139/1 Start, Action, Result up to 99 1139/2 Odd sums for odd jobs 1169/3 Renovating a house 1179/4 Planning our purchases 1169/5 Air freight 119

10 Adding, results beyond 100 12110/1 Start, Action, Result beyond 100 12110/2 Cycle camping 12110/3 One tonne van drivers 12210/4 Catalogue shopping 124

Num 3 Addition

36

Num 4 Subtraction

37

1 Actions on sets: taking away 2041/1 Start, Action, Result (do and say) 2041/2 Taking away on the number track (do and say) [NuSp 1.4/1] 205

2 Subtraction as a mathematical operation 2062/1 Predicting the result (subtraction) 2062/2 What will be left? (NuSp 1.4/2) 2072/3 Returning over the stepping stones 2072/4 Crossing back (NuSp 1.4/3) 208

3 Notation for subtraction: number sentences 2093/1 Number sentences for subtraction 2093/2 Predicting from number sentences (subtraction) 211

4 Number stories: abstracting number sentences 2124/1 Personalized number stories 2124/2 Abstracting number sentences 2134/3 Personalized number stories - predictive 214 5 Numerical comparison of two sets 2165/1 Capture (NuSp 1.4/4) 2165/2 Setting the table 2175/3 Diver and wincher 2175/4 Number comparison sentences 2195/5 Subtraction sentences for comparisons 221

6 Giving change 1256/1 Change by exchange 1256/2 Change by counting on 1266/3 Till receipts 126

7 Subtraction with all its meanings 1287/1 Using set diagrams for taking away 1287/2 Using set diagrams for comparison 1297/3 Usingsetdiagramsforfindingcomplements 1307/4 Using set diagrams for giving change 1307/5 Unpacking the parcel (subtraction) 131

8 Subtraction up to 20, including crossing the 10 boundary 1338/1 Subtracting from teens: choose your method 1338/2 Subtracting from teens: “Check!” 1358/3 Till receipts up to 20¢ 1358/4 Gift shop 136

9 Subtraction up to 99 1379/1 “Can we subtract?” 1389/2 Subtracting two-digit numbers 1399/3 Front window, rear window 1419/4 Front window, rear window - make your own 143

10 Subtraction up to 999 14410/1 Race from 500 to 0 14410/2 Subtracting three-digit numbers 14510/3 Airliner 14610/4 Candy store: selling and stocktaking 148

Num 4 Subtraction

38

Num 5 Multiplication

39

6 Building product tables: ready-for-use results 1616/1 Building sets of products 1616/2 “I know another way” 1636/3 Completing the products table 1646/4 Cards on the table 1666/5 Products practice 1666/6 Multiples rummy 168

7 Multiplying 2- or 3-digit numbers by single-digit numbers 1707/1 Using multiplication facts for larger numbers 1717/2 Multiplying 3-digit numbers 1727/3 Cargo boats 172

8 Multiplying by 10 and 100 1768/1 Multiplying by 10 or 100 1768/2 Explaining the shorthand 1788/3 Multiplying by hundreds and thousands 178

9 Multiplying by 20 to 90 and by 200 to 900 1799/1 “How many cubes in this brick?” (Alternative paths) 1799/2 Multiplying by n-ty and any hundred 180 10 Long multiplication 18210/1 Long multiplication 18210/2 Treasure chest 183

1 Actions on sets: combining actions 2421/1 Make a set. Make others which match 2431/2 Multiplying on a number track 2441/3 Giant strides on a number track 244

2 Multiplication as a mathematical operation 2482/1 “I predict - here” using rods 2482/2 Sets under our hands 249

3 Notation for multiplication: number sentences 2503/1 Number sentences for multiplication 2513/2 Predicting from number sentences 252

4 Number stories: abstracting number sentences 1514/1 Number stories (multiplication) 1514/2 Abstracting number sentences 1524/3 Number stories, and predicting from number sentences 153

5 Multiplication is commutative; alternative notations: binary multiplication 1555/1 Big Giant and Little Giant 1565/2 Little Giant explains why 1585/3 Binary multiplication 1595/4 Unpacking the parcel (binary multiplication) Alternative notations 159

Num 5 Multiplication

40

Num 6 Division

usingmultiplication resultsfor division

41

1 Grouping 1-2581/1 Start, Action, Result: grouping 1-2581/2 Predictive number sentences (grouping) 1-2601/3 Word problems (grouping) 1-260

2 Sharing equally 1-2642/1 Sharing equally 1-2642/2 “My share is . . .” 1-2662/3 “My share is . . . and I also know the remainder, which is . . .” 1-2662/4 Word problems (sharing) 1-268

3 Division as a mathematical operation 1873/1 Different questions, same answer. Why? 1873/2 Combining the number sentences 1903/3 Unpacking the parcel (division) 1903/4 Mr. Taylor’s game 191

4 Organizing into rectangles 1934/1 Constructing rectangular numbers [Same as Num 1.13/1] 1944/2 The rectangular numbers game [Same as Num 1.13/2] 195

5 Factoring: composite numbers and prime numbers 1965/1 Factors bingo 1975/2 Factors rummy 1975/3 Alias prime [Same as Num 1.14/1] 1985/4 The sieve of Eratosthenes [Same as Num 1.14/2] 199

6 Relation between multiplication and division 2016/1 Parcels within parcels 201

7 Using multiplication results for division 2047/1 A new use for the multiplication square 2047/2 Quotients and remainders 2057/3 VillagePostOffice 207

8 Dividing larger numbers 2098/1 “I’m thinking in hundreds . . .” 2098/2 “I’ll take over your remainder” 2108/3 Q and R ladders 2138/4 Cargo Airships 214

9 Division by calculator 2179/1 Number targets: division by calculator 217

Num 6 Division

42

Num 7 Fractions as a double operation as numbers as quotients

9Rounding decimal fractions in place value notation

43

1 Making equal parts 2191/1 Making equal parts 2191/2 Same kind, different shapes 2211/3 Parts and bits 2231/4 Sorting parts 2231/5 Match and mix: parts 225 2 Take a number of like parts 2272/1 Feeding the animals 2272/2 Traineekeepers,qualifiedkeepers 2292/3 Head keepers 230

3 Fractions as a double operation; notation 2323/1 Expanding the diagram 2333/2 “Please may I have?” (Diagrams and notation) 235

4 Simple equivalent fractions 2374/1 “Will this do instead?” 2374/2 Sorting equivalent fractions 2384/3 Match and mix: equivalent fractions 239

5 Decimal fractions and equivalents 2405/1 Making jewellery to order 2405/2 Equivalent fraction diagrams (decimal) 2425/3 Pair, and explain 2435/4 Match and mix: equivalent decimal fractions 244

6 Decimal fractions in place-value notation 2456/1 Reading headed columns in two ways 2456/2 Same number, or different? 2476/3 Claiming and naming 248

7 Fractions as numbers. Addition of decimal fractions in place-value notation 2507/1 Target, 1 2517/2 “How do we know that our method is still correct?” 252

8 Fractions as quotients 2548/1 Fractions for sharing 2558/2 Predict, then press 2578/3 “Are calculators clever?” 2588/4 Number targets by calculator 259

9 Rounding decimal fractions in place-value notation 2619/1 Rounding decimal fractions 2619/2 Fractional number targets 262

Num 7 Fractions

44 Space 1 Shape

45

1 Sorting three dimensional objects 1-2841/1 Sorting by shape 1-2841/2 Do they roll? Will they stack? 1-2852 Shapes from objects 1-2862/1 Matching objects to outlines 1-2863 Lines, straight and curved 1-2873/1 Drawing pictures with straight and curved lines 1-2873/2 “I have a straight/curved line, like . . .” 1-2883/3 “Please may I have . . . ?” (Straight and curved lines) 1-2884 Line figures, open and closed 1-2904/1 “Can they meet?” 1-2904/2 Escaping pig 1-2914/3 Pig puzzle 1-2934/4 Inside and outside 1-2935 Sorting and naming two dimensional shapes 1-2955/1 Sorting and naming geometric shapes 1-2955/2 Sorting and naming two dimensional figures 1-2965/3 I spy (shapes) 1-2965/4 Claim and name (shapes) 1-2966 Shapes from objects and objects from shapes 1-2986/1 “This reminds me of . . .” 1-2987 Naming of parts 1-2997/1 “I am touching . . .” (three dimensions) 1-2997/2 “Everyone touch . . .” (three dimensions) 1-3007/3 “I am pointing to . . . (two dimensions) 1-3007/4 “Everyone point to . . .” (two dimensions) 1-3007/5 “My pyramid has one square face . . .” 1-3017/6 Does its face fit? 1-301

8 Parallel lines, perpendicular lines 2638/1 “My rods are parallel/perpendicular” 2638/2 “All put your rods parallel/perpendicular to the big rod” 2648/3 Colouring pictures 264 9 Circles 2669/1 Circles in the environment 2669/2 Parts of a circle 2669/3 Circles and their parts in the environment 2679/4 Patterns with circles 268

10 Comparison of angles 27010/1 “All make an angle like mine” 27010/2 “Which angle is bigger?” 27110/3 Largest angle takes all 27110/4 Angles in the environment 272

11 Classification of angles 27311/1 Right angles, acute angles, obtuse angles 27311/2 Angle dominoes 27411/3 “Mine is the different kind” 27511/4 “Can’tcross,willfit,mustcross” 275

12 Classification of polygons 27812/1 Classifying polygons 27812/2 Polygon dominoes 27912/3 Match and mix: polygons 279

13 Polygons: congruence and similarity 28213/1 Congruent and similar polygons 28313/2 Sides of similar polygons 28413/3 Calculating lengths from similarities 285

14 Triangles: classification, congruence, similarity 28714/1 Classifying triangles 28714/2 Triangle dominoes 28814/3 Match and mix: triangles 28814/4 Congruent and similar triangles 289

15 Classification of quadrilaterals 29115/1 Classifying quadrilaterals 29215/2 Relations between quadrilaterals 29315/3 “And what else is this?” 29415/4 “I think you mean . . .” 294

16 Classification of geometric solids 29616/1 What must it have to be . . . ? 296

17 Inter-relations of plane shapes 29817/1 Triangles and polygons 29817/2 Circles and polygons 29917/3 “I can see” 30117/4 Triangles and larger shapes 302

18 Tessellations 30418/1 Tessellating regular polygons 30418/2 Tessellating other shapes 30518/3 Inventing tessellations 30618/4 Tessellating any quadrilateral 307

19 Drawing nets of geometric solids 30919/1 Drawing nets of geometric solids 30919/2 Recognizing solids from their nets 31119/3 Geometric nets in the supermarket and elsewhere 312

Space 1 Shape

46

Space 2 Symmetry and motion geometry

10Relation betweenreflections, rotations, and flips

6Directions in space:north, south, east, west,and the half points

4Lines, rays,line segments

7Angles asamount of turn;compass bearings

8Directions and locations

5Translations oftwo-dimensional figures(slides without rotation)

3Two kinds of movement:translation and rotation

1Reflections oftwo-dimensional figures

9Rotations oftwo-dimensional figures;rotational symmetry

2Reflection andline symmetry

Parallel lines(Space 1)

47

1 Reflections of two-dimensional figures 3131/1 Animals through the looking glass 3141/2 Animals two by two 3141/3 Drawing mirror images 315

2 Reflection and line symmetry 3172/1 Introduction to symmetry 3172/2 Collecting symmetries 318

3 Two kinds of movement: translation and rotation 3203/1 Walking to school 320

4 Lines, rays, line segments 3224/1 Talk like a Mathematician (lines, rays, line segments) 3224/2 True or false? 323

5 Translations of two-dimensional figures (slides without rotation) 3265/1 Sliding home in Flatland 3265/2 Constructing the results of slides 3295/3 Parallels by sliding 330

6 Directions in space: north, south, east, west, and the half points 3326/1 Compass directions 3326/2 Directions for words 3336/3 Island cruising 334

7 Angles as amount of turn; compass bearings 3377/1 Directions and angles 3387/2 Different name, same angle 3397/3 Words from compass bearings 3397/4 Escape to freedom 341

8 Directions and locations 3448/1 Introduction to back bearings 3448/2 Get back safely 3458/3 Where are we? 3478/4 True north and magnetic north 350

9 Rotations of two-dimensional figures 3529/1 Centre and angle of rotation 3529/2 Walking the planks 3539/3 Introduction to rotational symmetry 3559/4 Match and mix: rotational symmetries 3569/5 Match and mix (line symmetry) 357

10 Relation between reflections, rotations, and flips 358

Space 2 Symmetry and motion geometry

48

NuSp 1 The number track and the number line

correspondencebetweensize of numberand positionon track

49

1 Correspondence between size of number and position on track 3061/1 “I predict - here” on the number track 306

2 Correspondence between order of numbers and position on track 3092/1 Sequences on the number track 309

3 Adding on the number track 3113/1 Putting more on the number track (verbal) [Same as Num 3.1/2] 3113/2 “Where will it come?” (Same as Num 3.2/2) 3133/3 Crossing (Same as Num 3.2/4) 3133/4 “Where will it come?” (Through 10) (Same as Num 3.7/2) 315

4 Subtracting on the number track 3164/1 Taking away on the number track (verbal) [Same as Num 4.1/2] 3164/2 What will be left? (Same as Num 4.2/2) 3164/3 Crossing back (Same as Num 4.4/4) 3174/4 Capture (Same as Num 4.5/1) 317

5 Relation between adding and subtracting 3195/1 Slow bicycle race 3195/2 Ups and downs 320

6 Linear slide rule 3226/1 Add and check 3236/2 Adding past 20 323

7 Unit intervals: the number line 3597/1 Drawing the number line 3597/2 Sequences on the number line 3607/3 Where must the frog land? 3607/4 Hopping backwards 3607/5 Taking 3607/6 A race through a maze 361

8 Extrapolation of the number line Extrapolation of the counting numbers 3638/1 What number is this? (Single starter) 3638/2 What number is this? (Double starter) 3658/3 Is there a limit? 3668/4 Can you think of . . .? 366

9 Interpolation between pairs Fractional numbers (decimal) 3689/1 What number is this? (Decimal fractions) 3699/2 Snail race 3709/3 Snails and frogs 371

10 Extrapolation of place-value notation 37310/1 “How can we write this number?” (Headed columns) 37310/2 Introducing the decimal point 37410/3 Pointing and writing 37510/4 Shrinking and growing 376

NuSp 1 The number track and the number line

50

Patt 1 Patterns

51

1 Patterns with physical objects 1-3281/1 Copying patterns 1-3281/2 Patterns with a variety of objects 1-3291/3 Making patterns on paper 1-329

2 Symmetrical patterns made by folding and cutting 1-3312/1 Making paper mats 1-3312/2 Bowls, vases, and other objects 1-3322/3 Symmetrical or not symmetrical? 1-333

3 Predicting from patterns 1-3343/1 What comes next? 1-3343/2 Predicting from patterns on paper 1-334

4 Translating patterns into other embodiments 3814/1 Different objects, same pattern 3814/2 Patterns which match 3824/3 Patterns in sound 3824/4 Similarities and differences between patterns 3834/5 Alike because . . . and different because . . . 384

Patt 1 Patterns

52

Meas 1 Length

53

1 Measuring distance 1-3411/1 From counting to measuring 1-3421/2 Tricky Micky 1-3421/3 Different names for different kinds of distance 1-344

2 The transitive property; linked units 1-3462/1 Building a bridge 1-346

3 Conservation of length 1-3483/1 “Can I fool you?” (length) 1-3483/2 Grazing goat 1-349

4 International units: metre, centimetre 3854/1 The need for standard units 3854/2 Counting centimetres with a ruler 3864/3 Mountain road 3874/4 Decorating the classroom 389

5 Combining distances corresponds to adding numbers of units 3905/1 Model bridges 3905/2 How long will the frieze be? 391

6 Different sized units for different jobs: kilometre, millimetre, decimetre 3926/1 “Please be more exact” (Telephone shopping) 3926/2 Blind picture puzzle 3936/3 “That is too exact” (Power lines) 3946/4 Using the short lengths (Power lines) 3956/5 “That is too exact” (Car rental) 3956/6 Chairs in a row 396

7 Simple conversions 3977/1 Eqivalent measures, cm and mm 3977/2 Buyer, beware 3987/3 A computer-controlled train 399

8 The system overall 4028/1 Relating different units 4028/2 “Please may I have?” (Metre and related units) 403

Meas 1 Length

54

Meas 2 Area

55

Meas 2 Area

1 Measuring area tiling a surface counting tiles to measure a surface square centimetre as a standard unit centimetre grid as a measuring instrument 4051/1 “Will it, won’t it? 4051/2 Measuring by counting tiles 4061/3 Advantages and disadvantages 4061/4 Instant tiling 407

2 Irregular shapes which do not fit the grid 4082/1 Shapes and sizes 4082/2 “Hard to know until we measure” 4082/3 Gold rush 409

3 Rectangles (whole number dimensions) Measurement by calculation 4113/1 “I know a short cut” 4113/2 Claim and explain 4123/3 Carpeting with remnants 412

4 Other shapes made up of rectangles 4144/1 ‘Home improvement’ in a doll’s house 4144/2 Claim and explain (harder) 4154/3 Net of a box 4154/4 Tilingthefloorsinahome 416

5 Other shapes convertible to rectangles 4175/1 Area of a parallelogram 4175/2 Area of a triangle 4195/3 Area of a circle 421

6 Larger units for larger areas: square metre, hectare 4236/1 Calculating in square metres 4236/2 Rentingexhibitionfloorspace 4246/3 Buying grass seed for the children’s stately garden 4246/4 Calculating in hectares 4266/5 Buying smallholdings 426

7 Relations between units 4287/1 What could stand inside this? 4287/2 Completing the table 4287/3 “It has to be this one” 429

8 Mixed units 4318/1 Mixed units 431

56

Meas 3 Volume and Capacity

6standard units(litres, millilitres,and kilolitres)

57

1 Containers: full, empty 3571/1 Full or empty? 357

2 Volume: more, less 3582/1 Which is more? 358

3 Conservation of volume 3593/1 Is there the same amount? 3593/2 “Can I fool you?” 360

4 Capacity of a container: comparing capacities 4074/1 Which of these can hold more? 407

Meas 3 Volume and Capacity

5 Measuring volume and capacity using non-standard units 4335/1 Putting containers in order of capacity 4335/2 “Hard to tell without measuring” 433

6 Standard units (litres, millilitres, and kilolitres) 4346/1 Can you spot a litre? 4346/2 “Special: Small Lemonade, 10¢” 4356/3 Making and tasting (accuracy in the kitchen) 4356/4 Race to a litre 4366/5 Completing the table of units of capacity or volume 436

58 Meas 4 Mass and weight

59

1 Introductory discussion 363 No activity; but a note for teachers 363

2 Comparing masses: heavier, lighter (estimation) 3642/1 “Which one is heavier?” 364

3 Comparing masses: the balance 3653/1 “Which side will go down? Why?” 3653/2 Find a pair of equal mass 3663/3 Trading up 366

4 Measuring mass by weighing, non-standard units 4384/1 Problem: to put these objects in order of mass 4384/2 Honest Hetty and Friendly Fred 439

Meas 4 Mass and weight

5 Standard units (kilograms) 4415/1 Making a set of kilogram masses 4415/2 Mailing parcels 442

6 The spring balance 4436/1 Checking a spring balance 4436/2 Mailing parcels (spring balance) 443

7 Grams, tonnes 4447/1 “I estimate __ grams.” 4447/2 “How many grams to a litre? It all depends.” 4457/3 A portion of candy 4457/4 The largest animal ever 446

8 Relations between standard measures 4478/1 Completing the table of units of mass 447

60

Meas 5 Time

of time

61

1 Passage of time 3711/1 Thinking back, thinking ahead 372

2 Order in time 3732/1 Thinking about order of events 3732/2 Relating order in time to the ordinal numbers 374

3 Stretches of time and their order of length 3753/1 “If this is an hour, what would a minute look like?” 3753/2 “If this is a month, what would a week look like?” 3763/3 Winning time 3773/4 Time whist 377

4 Stretches of time: their order of occurrence 3784/1 Days of the week, in order 3784/2 Days acrostic 3794/3 Months of the year, in order 3794/4 Months acrostic 379

5 Stretches of time: their relative lengths 3815/1 Time sheets 3815/2 Thirty days hath November . . . 382

6 Locations in time: dates 4496/1 What the calendar tells us 4496/2 “How long it is . . . ?” (Same month) 4506/3 “How long is it . . .?” (Different month) 4506/4 “How long is it . . . ?” (Different year) 451

7 Locations in time: times of day 4527/1 “How do we know when to . . .?” 4527/2 “Quelle heure est-il?” 4537/3 Hours, halves, and quarters 4547/4 Time, place, occupation 454

8 Equivalent measures of time 4568/1 Pairing equivalent times 456

Meas 5 Time

62 Meas 6 Temperature

63

1 Temperature as another dimension 3911/1 Comparing temperatures 3911/2 “Which is the hotter?” 392

2 Measuring temperature by using a thermometer 4572/1 The need for a way of measuring temperature 4572/2 Using a thermometer 458

3 Temperature in relation to experience 4603/1 Everyday temperatures 4603/2 “What temperature would you expect?” 4613/3 Temperature in our experience 462

Meas 6 Temperature

64

65

THE NETWORKS AND ACTIVITIES

‘Slippery slope’ [Num 3.7/3, SAIL Volume 1] *

* A frame electronically extracted fromNUMERATION, ADDITION AND SUBTRACTION,

a colour videocassette produced by theDepartment of Communications Media,

The University of Calgary

66

Tens and hundreds of cubes [Org 1.13/1]

67

[Org 1] SET-bASED ORgANIzATION Organizing in ways which lay foundations for

concepts relating to number

Org 1.13 Base ten

Concept Tens, hundreds, thousands.

Abilities (i) To form and recognize sets of these numbers. (ii) To recognize and match different physical representations of them, embodying differ-

ent levels of abstraction.

Activity 1 Tens and hundreds of cubes [Org 1.13/1] A teacher-led activity for a small group. Its purpose is to help children to transfer to base

ten the same thinking as they have developed for bases three, four, and five in earlier top-ics.

Materials • A large box of 1 cm cubes.

What they do 1. Introduce this activity by asking if they know what base is most used in everyday life. Relate this to the fact that we have ten fingers.

2. Ask them to make some ten-rods. 3. Ask them to join these up to make ten-squares. 4. Very likely they will run out of cubes: certainly they will not be able to make many

ten-squares. This may surprise them! Tell them that the number of single cubes in a ten-square is called a hundred, which (as they have discovered) is quite a large number. Tell them that we’re now on our way to big numbers.

Discussionof concepts

Base 10 involves the same concepts as those used in bases 3, 4, 5. Practically,however, there are 3 major differences: (i) Our monetary system, and themeasures used in commerce, technology and sciences, all use base 10; (ii) Thisbase enables us to represent larger numbers with fewer figures; (iii) But 10 is toobig to subitize. So manipulations which can be done perceptually for bases up tofive depend on counting when working with base ten. Whether, using hindsight,it might have been better to choose some other base, may be for some aninteresting subject for discussion. Many of us can remember the mixture of bases4, 12, and 20 used in England's former monetary system. Our measurement of time stillstill uses bases 60, 12, 24, 7, 30, 31, and 365! Computers work in base 2. This is not anot a good one for humans, who convert to base 16 (hexadecimal). But for mostpractical and theoretical purposes, base ten is the established one, and it is animportant part of our job as teachers to help children acquire understanding,confidence, and fluency (in that order of importance) in working with base ten indecimal notation. It is with this aim that the foundations have been so carefullyprepared in topics 1 to 12 of this network.

68

Activity 2 Tens and hundreds of milk straws [Org 1.13/2]

An activity for a small group. Its purpose is to repeat grouping into tens and hun-dreds with a different embodiment.

Materials • A large number of milk straws cut into halves. • Rubber bands.

What they do 1. Ask the children to find out how many hundreds, tens, and ones there are. 2. Working together they group the straws first into tens with a rubber band around

each ten. 3. They then group the tens into bundles of ten tens, with a rubber band around

each big bundle.

Activity 3 Thousands [Org 1.13/3]

A teacher-led activity for a small group. Its purpose is to combine their concept of a base cubed with that of base 10 to give them the concept of a thousand.

What they do 1. Repeat Activity 1, but ask them now to go on to see if they can make a ten-cube. 2. They will not have enough cubes to do this. Explain that the number of single

cubes in a ten-cube is called a thousand; it is a very large number. 3. If we haven’t enough single cubes to make a ten-cube, what can we do instead? 4. One possibility is to make a hollow cube using a ten-square for base and ten-

rods for the other eight edges. We then have to imagine a solid cube, made from 10 ten-squares on top of each other.

5. Another possibility is to use the base 10 cubes from a multibase set. We then have to imagine all the cubes inside, indicated by the markings on the faces.

In the preceding topics, children developed the concepts of a base, and of rods, squares, and cubes, using bases small enough for all the grouping to be done physi-cally. With base ten, this becomes laborious; and when it comes to the cube of base ten, impractical. However, provided that the earlier concepts have been well established,they can be combined with children’s concept of the number ten in such a waythat the concepts which were learned by using bases three, four, five expand to include base ten. This is what we have been doing in this topic. We have thus gradually been reducing children’s dependence on physical objects forrepresenting numbers and moving them towards representing them in other ways. We have also been extending children’s concepts of numbers into the thousands.They have now come a long way, and should be allowed to return to the supportof physical materials at any time when they feel the need. This will help to keep their concepts of numbers strong and active, and reduce the danger that when sym-bols become the main method for handling numbers, the concepts fade away.

OBseRVe anD LIsten ReFLeCt DIsCUss

Org 1.13 Base ten (cont.)

Discussion of activities

69

[Num 1] Numbers aNd their properties Numbers as mental objects which, like physical objects, have particular

properties

Num 1.11 EXTRAPOLATION OF NUMBER CONCEPTS TO 100

Concepts the complete counting numbers in order to 100, grouped in tens.

Abilities (i) to state the number of a given set from 1 to 100. (ii) to make a set of a given number from 1 to 100.

here we are concerned, not with a totally new concept (such as being odd or even), but with increasing the examples which a child has of his existing concept of number. order and completeness provide a framework to ensure that these new examples fit the established pattern. the key feature of the extrapolation is the idea that we can apply the process of counting, not only to single objects but to groups of objects, treating each group as an entity. so this topic links with all the topics in org 1, ‘set-based organization,’ which are shown in the upper part of the network as leading to topic 13 (org 1.13, ‘base ten’).

activity 1 Throwing for a target [Num 1.11/1] an activity for one player, two working together, or it may be played as a race be-

tween two players throwing alternately. its purpose is to help children to extrapolate their number concepts up to 100, and to consolidate their use of grouping in tens.

Note before this activity, children should have completed org 1.

Materials • a game board, see Figure 1.* • For stage (a), base 10 material. • For stage (b), a variety of other materials as described below. • 2 dice. • slips of paper on which are written target numbers, e.g., 137, 285. (it is best not

to go above 300 at most, or the game takes too long.) * provided in photomasters

What they do Stage (a) 1. the player throws the dice and adds the two numbers. 2. he puts down that number of ones on the game board, one per line and starting

from the top. 3. each time he reaches 10 ones he exchanges them for one 10, placing it on a line

in the ‘tens’ column. similarly, 10 tens are exchanged for one 100 to go in the hundreds’ column.

4. He must finish with a throw of the exact number to reach the target number. 5. if the number is 6 or less he uses one die only.

discussion of concept

70

Num 1.11 Extrapolation of number concepts to 100 (cont.)

tens ones

ones ones

hundreds

Now you may exchangeten tens for one hundred.

Now you may exchangeten ones for one ten.

Figure 1 Throwing for a target

Stage (b) to prevent children becoming too attached to a particular embodiment, this game

should also be played with other suitable materials, such as popsicle sticks for ones, bundles of ten with a rubber band around, and ten bundles of ten. drinking straws cut in half are good. also: pennies, dimes, and $1 coins.

Notes (i) at stage (a), children will

often put more than ten ones in the ‘ones’ column, and these should be available. e.g., starting with the state on the left, and throwing 5, they put down 5 and get the state on the right. then they remove the line of ten cubes, exchanging these for a ten-rod which they put in the ten column. this is a good way to begin.

(ii) however, at stage (b) provide only 10 ones. this leads to a variety of strategies, which children should be given time to discover for themselves. it is important to restrain one’s urge to tell them, or they may acquire the method but not its interior-ity. I recommend that you say something like this: “When you find short cuts, as soon as you are sure they are correct you may use them.”

71

Num 1.11 Extrapolation of number concepts to 100 (cont.)

discussion of activities

activity 2 Putting and taking [Num 1.11/2] a game for two players. it is a simple variant of activity 1, with the same purpose.

Materials • 50 (halved) milk straws in bundles of ten, for each player. • ‘exchange pool’: 5 more tens and 20 ones. • 2 dice. • a box to put spares in.

What they do 1. each player starts with 50 straws, in bundles of 10. they agree which will put, and which will take, each working separately with his own set of straws.

2. they throw the dice alternately. 3. the ‘putter’ begins and puts down at his place the number that is the total shown

on the dice. 4. the ‘taker’ plays next and takes away from the bundles at his place the total

shown when he throws the dice. 5. the ‘putter’ wins by reaching 100. 6. the ‘taker’ wins by reaching 0. 7. exchanging of 1 ten for 10 ones will be necessary whenever they cross a ten

boundary upwards or downwards. 8. if the game is played with the same requirement as in activity 1, that the exact

number must be thrown to win, this could prolong the game unduly. it is there-fore probably best to agree that a throw which would take the number past 100 or 0 is also acceptable.

activities 1 and 2 both use the now-familiar concepts of grouping in tens, and ca-nonical form (see org 1.12) to lead children on to 100. although they are extrapo-lating their number concepts, which is schema building by mode 3 (creativity), this extrapolation is also strongly based on physical experience (mode 1 building). this is an excellent combination. an interesting feature here is that the physical experience by itself would not be sufficient to lead to the formation of these new concepts. A suitable schema is also needed which can organize this experience, and contains a pattern ready to be extrapolated by this experience. this has never been put better than by Louis pasteur, when he said: “discoveries come to the prepared mind.”

OBSERVE AND LISTEN REFLECT DISCUSS

72

Num 1.12 ORDINAL NUMBERS, FIRST TO ONE hUNDREDTh

Concepts The ordinal numbers, first to one hundredth.

Abilities For any set of objects which have an order: (i) to identify the ordinal number of a given object; (ii) Given an ordinal number, to find the object to which it belongs.

the children should already have the concept of an ordinal number, with examples from first to tenth, so the present topic only requires the expansion of a pattern which they already know. this pattern is regular, apart from the teens; and even there, the derivation from the counting numbers is regular. the task is therefore a straightforward one, and it is likely that a number of children will already have learned it without having been specifically taught.

activity 1 Continuing the pattern [Num 1.12/1]

a straightforward activity for a small group of children. its purpose is to extend the pattern of ordinal numbers up to hundredth.

Materials None.

What they do 1. They begin by saying the ordinal numbers from first to tenth, in order clockwise around the table.

2. “Now we go on in tens, like this. tenth, twentieth, thirtieth, — can you go on?” The children continue in turn “fortieth, fiftieth, . . . hundredth.”

3. if necessary, this is practised until the children can do it reliably. 4. “Now we put in the numbers in between.” starting at tenth, the children con-

tinue round the table, initially with prompting from yourself, “eleventh, twelfth, thirteenth, . . . twentieth, twenty-first, twenty-second, . . . long enough to demon-strate that they have grasped the pattern.

5. they should then be able to continue from any starting point, taking turns to begin with any ordinal number they like (below hundredth). the others then continue round the table. this continues until they can do it reliably, from any chosen starting point.

activity 2 sorting proverbs [Num 1.12/2]

an activity for children to do individually. its purpose is to provide an application of the concepts introduced in Activity 1. This activity is not conceptually difficult, but requires concentration and accuracy in the use of ordinal numbers.

Materials For each child • a copy of WiNdiNG Words (photomaster 6).* • one of the four cards provided on photomaster 7.* • pencil, paper, and eraser. * provided in SAIL Volume 2a.

discussion of concepts

73

What they do 1. each card has on it four proverbs, in which the words have been replaced by their ordinal numbers in the winding assortment of words shown in WiNdiNG WORDS. By using these numbers to find the corresponding words, each sepa-rate proverb can be sorted out from the mixture.

2. to help organize the process, some of the words are printed in bold. the chil-dren may be left to find out for themselves what this indicates. They should also be told to write the words on a single line, like a sentence.

3. What they must not do is to write anything on any of the cards. this would make things easier, but the object of this activity is to get lots of practice in using ordinal numbers. it would also spoil the cards for the next users.

4. it is desirable that the children do not see each other’s answers for the cards they have not yet done. one way to avoid this would be to have them all working on the same card, which would of course require extra copies.

5. it may be necessary to call attention to the difference between pairs like ‘thirti-eth’ and ‘thirteenth.’

6. some of the results may be somewhat unexpected! i suggest that you indicate which words are incorrect, and leave it for the children to put them right.

activity 1 is simple, but it embodies a principle of great generality in mathematics: that of extending a known pattern. in activity 2, i have taken the ordinals a little way beyond one hundredth, and have left this for the children to follow unassisted. having disentangled the proverbs, children may like to discuss their meanings, with your help. paraphrasing is not always easy, since the essence of a proverb is that it says something in a way which is catchy and concise. the meaning may be more easily conveyed by examples. this discussion may also help to expand children’s knowledge in other areas. For example, proverb 8 refers to migration; proverb 6, to good housekeeping (and it may also save losing a button). there may even be children who do not know how an omelette is made! i thought it more appropriate to leave proverb 14 as used in its country of origin, so an elementary knowledge of foreign currency is required. a list of the proverbs is appended, for your own information.

1. there’s many a slip twixt cup and lip. 2. he who laughs last laughs loudest. 3. Birds of a feather flock together. 4. time and tide wait for no man. 5. all that glisters is not gold. 6. a stitch in time saves nine. 7. don’t look a gift horse in the mouth. 8. one swallow doesn’t make a summer. 9. You can’t make an omelette without breaking eggs. 10. half a loaf is better than no bread. 11. Least said soonest mended. 12. every cloud has a silver lining. 13. darkest the night when dawn is nigh. 14. in for a penny in for a pound. 15. it’s love that makes the world go round. 16. it’s an ill wind that blows no one any good.

discussion of activities

Num 1.12 Ordinal numbers first to one hundredth (cont.)

OBSERVE AND LISTEN REFLECT DISCUSS

74

discussion of concepts

Num 1.13 RECTANgULAR NUMBERS

Concept rectangular numbers, as the number of unit dots in a rectangular array. (see exam-ple below.)

Ability to recognize and construct rectangular numbers.

here is another property which a number can have, or not have. the term ‘rectan-gular’ describes a geometrical shape, so when it is applied to a number it is being used metaphorically. provided we know this, it is a useful metaphor, since the cor-respondences between geometry and arithmetic (also algebra) are of great impor-tance in mathematics. rectangular numbers are closely connected with multiplication and with calcu-lating areas. they provide a useful contribution to both of these concepts, and a connection between them.

activity 1 Constructing rectangular numbers [Num 1.13/1] an activity for a small group, working in pairs. its purpose is to build the concept of

rectangular numbers.

Materials For each pair: • 25 small counters with dots at their centres. • pencil and paper.

What they do 1. the activity is introduced along the following lines. explain, “We think of these counters as dots which we can move around.”

2. put out a rectangular array such as this one.

3. ask, “What shape have we made?” (a rectangle.) 4. “how many counters?” (in this case, 12.) 5. “so we call 12 a rectangular number.” 6. repeat, with other examples, until the children have grasped the concept. Note

that the rectangles must be solid arrays like the one illustrated. 7. Next, let the children work in pairs. Give 25 counters to each pair, and ask them

to find all the rectangular numbers up to 25. 8. if any of them think that a pattern like this might be a rectangle, remind them that the counters represent dots, draw these on paper, and ask: “Would you say this is a rectangle?”

75

9. the question always arises whether or not squares are to be included. the an-swer is “yes.” both squares and oblongs are rectangles, in the same way as both girls and boys are children. this is discussed fully in space 1.15.

10. Finally, let the children compare and check their lists with each other’s.

activity 2 The rectangular numbers game [Num 1.13/2] a game for two. it consolidates the concept of a rectangular number in a predictive

situation. Children also discover prime numbers, though usually they do not yet know this name for them.

Materials • 25 counters. • pencil and paper for scoring.

1. each player in turn gives the other a number of counters. 2. if the receiving player can make a rectangle with these, he scores a point. if not,

the other scores a point. 3. if when the receiving player has made a rectangle (and scored a point), the giv-

ing player can make a different rectangle with the same counters, he too scores a point. (e.g., 12, 16, 18).

4. the same number may not be used twice. to keep track of this, the numbers 1 to 25 are written at the bottom of the score sheet and crossed out as used.

5. the winner is the player who scores the most points.

activity 1 uses physical experience for building the concept of a rectangular number. Communication is also used to introduce the concept and to attach the ac-cepted name. schema building thus takes place by modes 1 and 2 in combination. activity 2 uses rectangular numbers as a shared schema which forms the basis of a game. success at this game depends on predicting whether or not a given number is rectangular. each prediction is tested (mode 1) as part of the game.

OBSERVE AND LISTEN REFLECT DISCUSS

Rules of the game

Num 1.13 Rectangular Numbers (cont.)

discussion of activities

76

Num 1.14 Primes

Concepts (i) a prime number as one which is not a rectangular number. (ii) a prime number as one which is not the multiple of any other number except 1

and itself. (iii) a prime number as one which is not divisible by any other number except 1 and

itself.

Abilities (i) to use these criteria to recognize prime numbers. (ii) to give examples of prime numbers. (iii) to be able to construct the set of primes less than 100.

this is a negative property, that of not having a given property. Children will already have formed this concept while playing the rectangular numbers game. in this topic we give the concept further meaning by relating it to other mathematical ideas.

activity 1 Alias prime [Num 1.14/1]

a game for up to 6 players. its purpose is to introduce children to the difference between composite and prime numbers, and give them practice in distinguishing between these two kinds of number.

Materials • three counters for each player.

1. begin by explaining the meanings of ‘composite number’ and ‘prime.’ these concepts have been well prepared in earlier activities, and children have usually invented their own names for them.

2. explain that ‘alias’ means ‘another name for,’ often used to hide someone’s iden-tity. in this game, all prime numbers use the alias ‘prime’ instead of their usual name.

3. start by having the players say in turn “eight,” “Nine,” “ten,” . . . around the table.

4. the game now begins. they say the numbers around the table as before, but when it is a player’s turn to say any prime number, they must not say its usual name, but say “prime” instead.

5. the next player must remember the number which wasn’t spoken, and say the next one. thus the game would begin (assuming no mistake) “eight,” “Nine,” “ten,” “prime,” “twelve,” “prime,” “Fourteen,” “Fifteen,” “sixteen,” “prime,” “eighteen,” and so on.

6. any player who makes a mistake loses a life — i.e., one of her counters. Failing to say “prime,” or saying the wrong composite number, are both mistakes.

7. When a player has lost all her lives she is out of the game, and acts as an umpire. 8. the winner is the last player to be left in the game.

Note When the players are experienced, they may begin counting at “one.” this gives rather a lot of primes for beginners!

discussion of concepts

Rules of the game

77

activity 2 the sieve of eratosthenes [Num 1.14/2]

an activity for children working individually. as they get better at ‘alias prime,’ they will come to numbers about which they are doubtful whether these are prime or not. this is a method for constructing a set of primes by systematically eliminating multiples.

Materials • For each child a 10 by 10 number square on which are written the numbers in or-der from 1 to 100. they can make these themselves, but two copies are included in the Volume 2a photomasters.

What they do 1. they begin by crossing off all the multiples of 2 except 2 itself. 2. then they cross out all the multiples of 3 except 3 itself, and so on. 3. After 7, they will find that there are no new ones to cross out. Let them discover

this. Why is it so? (this is quite a hard question.) 4. When finished, children check their results against each other’s. 5. they then make a separate list of the primes below 100.

activity 3 Sum of two primes [Num 1.14/3]

a game for two. its purpose is to give a further opportunity for thinking about prime numbers.

Materials • Number cards 2 to 20.* * provided in the photomasters

What they do Stage (a) 1. The cards are shuffled and put on the table face down. 2. player a turns over the top card and puts it down face up. (say, 14.) 3. she now tries to express this as the sum of two primes, and if successful she

scores a point. in this case she can: 13 + 1. 4. if b can do so in a different way, she also scores a point. in this case she can: 3 + 11. 5. b then turns over the next face-down card, and the game is continued as above.

Stage (b) this is played as above, but using two dice to give larger numbers. these are of dif-

ferent colours, and one determines the tens, the other the ones.

Num 1.14 Primes (cont.)

78

the concept of a prime number was implicit in Num 1.13/2, ‘the rectangular numbers game,’ when children encountered numbers which are not rectangular numbers. here this negative property is made explicit and given a name. activity 1 (‘alias prime’) centres attention on prime numbers in a game based on this con-cept, and the concept is tested by mode 2 (comparison with the schemas of others, leading sometimes to discussion). primes were initially conceptualized in relation to rectangular numbers, which is a physical and spatial metaphor. in activity 2 they are thought of in a different way, that of not being multiples. they are therefore not divisible except by 1 and themselves. multiplication and division are abstract mathematical operations, so the concept of a prime has now become independent of its physical/spatial begin-nings. independent does not however mean permanently detached from physical embodiments. While learning to climb a mountain higher and higher, we must always retain the ability to come down again. We also need to be able to put our mathematics to work, and this means relating it once again to physical embodi-ments. activity 3 is an abstract game with numbers, of a kind which only mathemati-cians enjoy. so if the children we are teaching do enjoy this game, we may con-gratulate both them and ourselves.

discussion of activities

Num 1.14 Primes (cont.)

OBSERVE AND LISTEN REFLECT DISCUSS

Square numbers [Num 1.15/1]

79

Num 1.15 Square numbers

Concept a square number as a particular kind of rectangular number, in which the corre-sponding rectangle has sides equal in length (i.e., is a square).

Ability to construct and recognize square numbers.

since a square is a particular kind of rectangle, we are now distinguishing a partic-ular kind of rectangular number, using the term ‘square’ metaphorically as was the case for ‘rectangle.’ Note that ‘a three-square’ is a kind of square, whereas ‘three squared’ is a number: the number of dots in a three-square. the latter is the result of an operation on the number 3 (multiplied by itself); compare 3 doubled and 3 halved.

activity 1 Square numbers [Num 1.15/1]

an activity for a small group. its purpose is to introduce the concept of a square number.

Materials • 50 counters. • pencil and paper for each child.

What they do 1. ask the children to make the rectangles for the numbers 4, 9, 16. 2. put these in the centre of the table, and ask the children what they notice. 3. explain that since these rectangles are squares, their numbers also are called

squares. 4. ask the children each to make a list of all the square numbers up to 100. they

may use counters if they wish, and if there are enough. 5. When they have finished, they compare lists. If there is any discrepancy, it is

discussed.

activity 2 An odd property of square numbers [Num 1.15/2] this is an interesting investigation which can be done singly or in pairs. it relates

the sequence of square numbers (present topic) and of odd numbers (Num 1.9 in Volume 1).

Materials • Cubes: about 30 in each of several colours. • paper and pencil.

What they do 1. ask the children to continue the following, each time adding the next odd number, until they have seen a pattern.

1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16

discussion of concept

80

2. When they have seen the pattern, ask them to try and find out why it is that add-ing the next odd number gives the next square number. This is difficult without the following hint: make a staircase of the odd numbers, using different colours for adjacent rods. the step from this to an explanation is shown in the diagram below; but if children can discover this for themselves, it is a pity to deprive them of this pleasure.

3. A further step is to see that the sum of the first N odd numbers is equal to the square of N. This is difficult to express tidily in words without the use of N to stand for ‘any number.’

etc. makingThenthis.

Andso on.

hiNt eXpLaNatioN

Num 1.15 Square numbers (cont.)

First this . . .

An odd property of square numbers [Num 1.15/2]

81

activity 1 shows a familiar pattern. a new concept is introduced by examples of several physical embodiments. Children then use the concept to find more exam-ples for themselves, first with and then without the help of physical embodiments. all three modes of schema building are thus brought into use. activity 2 introduces not just a property of certain numbers, but a relation between two patterns. one pattern is formed by the successive left-hand sides, the other by the successive right-hand sides of the equations. this result can also be obtained starting from the other end, i.e. beginning with the explanation shown after step 2, and deriving the sequences of numbers on the left and right-hand sides. but i think that the way i have presented it makes it more intriguing, and increases the pleasure of discovering why.

here is a question which has come up a number of times in discussion with teachers. as explained in the discussion of the concept, 3 multiplied by itself is called 3 squared.

so 2 squared = 4 3 squared = 9 4 squared = 16 and so on.

the inverse relationship is called a square root.

so the square root of 4 = 2 the square root of 9 = 3 the square root of 16 = 4 and so on.

as long as we are talking about the counting numbers, only certain numbers have a square root. these are called perfect squares. so 4, 9, 16 . . . are perfect squares. they are also non-prime. are all perfect squares non-prime? in particular what about 1? You might like to think about this, and perhaps discuss it with others, before reading on. the discussion is probably more important than the result. here are my own thoughts on the matter. 1 is not a square number, since a single dot can hardly be described as a square. however, 1 multiplied by itself is equal to 1, i.e., 12 is 1. so we now have the paradox that 1 is a perfect square arithmetically, but geometrically it is not a square number. By any definition, 1 is prime.

in this context, 1 is unique.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 1.15 Square numbers (cont.)

1

23

4

5etc.

9

etc.7

primes perfect squares

For your own interest

discussion of activities

82

Num 1.16 Relations between numbers

Concept that all numbers are related, in many ways.

Abilities (i) To find several relationships between a given number and other numbers. (ii) To find several relationships between two given numbers.

From the second topic of this network onwards, there has been emphasis not only on individual numbers but on relationships between them. this is a particular case of one of the major emphases of this approach: that learning mathematics involves learning, not isolated facts, but a connected knowledge structure. The relationships dealt with specifically in this topic are straightforward ones such as successor and predecessor, half and double. in other networks we meet (e.g.) sum and difference, multiple and factor. in this topic we go further, to bring forward a very general property of numbers: that every number is related to every other number, not just in one way but many. indeed, the only limits to how many relationships we can find are those of time and patience.

activity 1 “Tell us something new.” [Num 1.16/1]

a game for a small group. its purpose is to give practice in thinking about properties of numbers, and their relationships with other numbers.

Materials • a bowl of counters, say 3 per player. • a pack of number cards (or any other way of generating assorted numbers).

how high they should go depends on the ability of the players.

1. The first card is turned over, e.g., 17. 2. each player in turn has to say something new about this number, e.g.,

“17 is prime.” “17 is 10 + 7.” “17 is 20 - 3.” “17 is half 34.” “17 is four fours plus 1.” etc. 3. each time a player makes a correct statement, the others say “agree” and she

takes a counter. 4. if it has been used already, the others say, “tell us something new.” 5. if an incorrect statement is made, they say, “tell us something true.” 6. When all the counters have been taken, many different properties and relation-

ships have been stated about the same number. 7. the player with the most counters is the winner. 8. another game may then be played with a different number.

Rules of the game

Discussion of concept

83

activity 2 “How are these related?” [Num 1.16/2]

a game for a small group. it is a variation of activity 1, but harder.

Materials • as for activity 1.

as for activity 1, except that two numbers are used at a time. players now have to say how these are related.

e.g., 25 and 7 “25 is more than 7.” “25 is 8 more than 7.” “3 sevens plus 4 makes 25.” “both 25 and 7 are odd numbers.” “the sum of 25 and 7 is an even number.” etc.

These activities develop fluency and inventiveness in the handling of numbers. In activity 1, children can choose to keep well within their domain (the mental region where they feel confident), or to stretch their abilities by devising more difficult an-swers and thereby to expand their domains. this is under each child’s control, so confidence is maintained and developed further. Activity 2 allows less choice, so it is more demanding. they also increase the interconnections within children’s schemas. this makes much use of Mode 3 activity – the creative use of existing knowledge to find new relationships. testing is by mode 2, agreement and if necessary discussion, which in turn is based on mode 3 – testing by consistency with what is already known.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 1.16 Relations between numbers (cont.)

Rules of the game

Discussion of activities

“How are these related?” [Num 1.16/2]

84

‘Number Targets’ [Num 2.8/1] *

* a frame electronically extracted fromNumeratioN, additioN aNd subtraCtioN,

a colour videocassette produced by thedepartment of Communications media,

the university of Calgary

85

[Num 2] The naming of numbers in ways which help us to organize them and use them

effectively

Num 2.8 WritteN Numerals 20 to 99 usiNg headed columNs

Concept That a particular digit can represent a number of ones, tens, (and later hundreds . . .) according to where it is written.

Abilities (i) To match numerals of more than one digit with physical representations of ones, tens, (and later hundreds . . .).

(ii) To speak the corresponding number‑words for numbers 20 to 99.

First, let us be clear about what a digit is. It is any of the single‑figure numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (corresponding to the numbers we can count on our fingers). Just as we can have words of one letter (such as a), two letters (such as an), three letters (such as ant), and more, so also we can have written numerals of one digit (such as 7), two digits (such as 72), three digits (such as 702), and more. The same numeral, say 3, can be used to represent 3 buttons, or shells, or cubes, or single objects of any kind. If we want to show which objects, we can do so in two ways. We can either write ‘3 buttons, 5 sea shells, and 8 cubes,’ or we can tabulate:

buttons sea shells cubes 3 5 8

Likewise the same numeral, say 3, can be used to represent 3 single objects, or three groups of ten, or 3 groups of ten groups of ten (which we call hundreds for short). We could write ‘3 hundreds, 5 tens, and 8 ones’; or we could tabulate:

hundreds tens ones 3 5 8

We are so used to thinking about (e.g.) 3 hundreds that we tend not to realize what a major step has been taken in doing this. We are first regarding a group of ten objects as a single entity, so that if we have several of these we can count “One, two, three, four, five . . . groups of ten.” Then we are regarding a group of ten groups of ten as another entity, which can likewise be counted “One, two, three . . . .” And by the end of this network, we shall no longer be regarding these as groups of physical objects, but as abstract mental entities which we can arrange and rear‑range. We shall also have introduced a condensed and abstract notation (place value). These two steps need to be taken one at a time. While the first, described above, is being taken, we need to use a notation which states clearly and explicitly what is meant: i.e., headed column notation. Also, because the correspondence between written numerals and number words only becomes regular from 20 onwards, we start children’s thinking about written numerals here where the pattern is clear. The written numerals 11 ‑ 19 are also regular, but their spoken words are not, so these are postponed until the next topic.

Discussion of concept

86

Num 2.8 Written numerals 20 to 99 using headed columns (cont.) activity 1 number targets [Num 2.8/1] A game for as many children as can sit so that they can all see the chart right way up;

minimum 3. It follows on from ‘Tens and ones chart’ (Num 2.7/3 in SAIL Volume 1). Its purpose is to link the spoken number words, just learned, with the corresponding written numerals.

Materials • Target cards.* • Tens and ones chart.* • Pencil and headed paper for each child. • Base 10 material, tens and ones.** * Provided in the photomasters. See also Note (iii), following. ** This game should be played with a variety of base ten material such as milk

straws or popsicle sticks in ones and bundles of ten; multibase material in base ten.

What they do 1. The target cards are shuffled and put face down. 2. In turn, each child takes the top card from the pile. He looks at this, but does not

let the others see it. 3. Before play begins, 2 tens are put onto the chart. (This is to start the game at

20.)

4. The objective of each player is to have on the chart his target number of tens and ones.

5. Each player in turn may put in or take out a ten or a one. 6. Having done this, he writes on his paper the corresponding numerals and speaks

them aloud in two ways. For example:

Tens Ones

Tens Ones

Target cards Paper

Tens Ones

4

5

7

4

87

Num 2.8 Written numerals 20 to 99 using headed columns (cont.) 7. In the above example, if a player holding a 47 target card had the next turn, he

would win by putting down one more one. He would then show his target card to confirm that he had achieved his target.

8. Since players do not know each others’ targets, they may unknowingly achieve someone else’s target for them. In this case the lucky player may immediately reveal his target card, whether it is his turn next or not.

9. When a player has achieved a target, he then takes a new target card from the top of the pile, and play continues.

10. The winner is the player who finishes with the most target cards.

Notes (i) If one side of the tray is empty, a corresponding zero must be written and spoken; e.g.,

(ii) Players are only required to write the numbers they themselves make. It would be good practice for them to write every number, but we have found that chil‑dren consider this to be cumbersome.

(iii) Several sets of target cards should be prepared in which the numbers are rea‑sonably close together, both the tens and the ones. If they are too far apart, the game may never end.

Variation It makes the game more interesting if, at step 5, a player is allowed two moves. For example, he may put 2 tens, or put 2 ones, or put 1 ten and take 1 one, etc. This may also be used if no one is able to reach his target.

activity 2 number targets beyond 100 [Num 2.8/2] When children are familiar with numbers greater than 100, they can play this game

with suitable modifications using targets in hundreds, tens and ones.* * Provided in the photomasters

In preparation for place‑value notation, it is important for children to have plenty of practice in associating the written symbols and their locations with visible em‑bodiments of hundreds, tens, and ones . . . as well as associating both of these with the spoken words. In this topic ‘location’ means ‘headed column’; in Num 2.10 it will mean ‘relative position.’ This activity uses concept building by physical experience (Mode 1). The social context provided by a game links these concepts with communication (Mode 2) using both written and spoken symbols.

oBserVe aNd listeN reFlect discuss

Discussion of activities

88

Discussion of concepts

Tens Ones

Num 2.9 WritteN Numerals From 11 to 20

Concepts The written numerals 11 ‑ 19 as having the same meanings as the number‑words with which they are already familiar.

Abilities (i) To match the numerals 11 ‑ 19 with the spoken number‑words. (ii) To match both with physical embodiments of these numbers.

The discussion of Num 2.8 applies equally here. However, the clear and regular correspondence which we find from 20 onwards, e.g.:

2 tens, 7 ones twenty seven 27

does not apply from 11‑19. E.g., although 1 ten, 2 ones is written (as we would expect) 12, it is spoken not as “onety two” but as “twelve.” And 1 ten, 7 ones is written (as we would expect) 17, but the spoken form is backwards, seventeen. So the present topic contains, implicitly, the notion of irregularity – departure from an expected pattern.

activity 1 seeing, speaking, writing 11‑19 [Num 2.9/1] A teacher‑led activity for a small group. Its purpose is to relate the number‑words

spoken but not written down in Num 1.7/1 to the written numerals 11‑19.

Materials • Tens and ones chart. • Base 10 material. • Paper and pencil for yourself. Suggested 1. You put in sequence for the discussion

2. Write

89

3. Say: “One ten, zero ones. Ten.” 4. Put in

5. Write

6. Say: “One ten, one one. Eleven.” 7. Carry on through the teens. (So you don’t put in another ten.) 8. Soon the children will join in. You might point out that if these followed the

same pattern as 21, 31 . . . we would talk about onety‑one, onety‑two, etc., and explain that in olden times people hadn’t thought about it carefully: so the ‘teen’ names came to be attached, and have ‘stuck.’

activity 2 number targets in the teens [Num 2.9/2] The same number targets game as in the topic just before this (Num 2.8/1) should

now be played, starting with the tens and ones chart empty.

Materials As for (Num 2.8/1) except: • Target cards now 11 to 19 (provided in the photomasters).

The written numerals 10 ‑ 19 follow the same pattern as those from 20 on, so these concepts are acquired by extrapolation (Mode 3 schema building). However, the spoken number‑words do not follow this pattern, and do not correspond well to the numerals, even from 13 on. (“Thirteen,” “fourteen” . . . are read from right to left.) However, children will already be familiar with the spoken number‑words, and the activity now links these to the numbers in physical embodiments (tens and ones chart), and to the written numerals.

oBserVe aNd listeN reFlect discuss

Tens Ones

Num 2.9 Written numerals from 11 to 20 (cont.)

Discussion of activities

90

Num 2.10 Place‑Value NotatioN

Concept That a particular digit can represent a number of ones, tens, hundreds, according to whether it comes first, second, third in order reading from right to left.

Abilities (i) To identify separately which digits represent ones, tens, hundreds, by their posi‑tions relative to each other.

(ii) To read aloud two and three digit numerals. (iii) To match these with physical representations.

Note Hundreds are not included until the second time around, after Num 2.13, ‘Nu‑merals beyond 100, written and spoken.’

Provided that we have only one digit in each column (which may be a zero), we can leave out the headings and ruled columns and still know what each digit stands for. The result is a brilliantly simple notation, whose brilliance is easily over‑looked just because it is so simple. The Greek mathematicians, excellent though they were in many ways, did not think of it; nor did the Romans, nor the earlier Egyptians nor the Babylonians. As a result, they made relatively slow progress in arithmetic and algebra. It is also a condensed and abstract notation, and this is why it has been ap‑proached with such careful preparation in the preceding activities.

activity 1 “We don’t need headings any more.” [Num 2.10/1] A teacher‑led activity for a small group. Its purpose is to introduce children to

place‑value notation, as described above.

Materials • Pencil and paper.

Suggested 1. Write some number between 21 and 99, with headed columns, as below. sequence for the discussion

2. Ask, “Can you say this number?” Accept either “forty‑seven” or “four tens, seven ones.”

3. Then say, “Can you say it another way?” to get the alternative reading. (At this stage we want both, every time.)

4. Repeat with other numbers, including ones between 11 and 19. Numbers such as 30 should be read as “Three tens, zero ones; thirty.”

Discussion of concept

91

5. Fold the top of the paper under so that the headings for tens and ones do not show, and write another number. Ask the children if they can still say the num‑bers as before. Practise this until they are confident, and then do the same with‑out the dividing line.

6. If the children are fully proficient in the new notation, say, “So we don’t always need the headings now, though they are still useful sometimes,” and continue to Activity 3. Children who have progressed through the earlier topics in this network should have no difficulty at this stage. For those who do, it would be best to go back to topics Num 2.8 and 2.9 to ensure that they are fully prepared. They should then do Activity 2 of this topic.

activity 2 number targets using place‑value notation [Num 2.10/2] The children should now play again the number targets game, exactly as in Num

2.8/1 and Num 2.9/1, but with plain number cards* and plain paper (no headings). * Provided in the photomasters.

activity 3 Place‑value bingo [Num 2.10/3] A game for 5 or 6 players. Its purpose is to consolidate children’s understanding of

the relationships between written numerals in place value notation, the same read aloud, and physical embodiments of the numbers.

Materials • Number cards 0 to 59.* • Base 10 material, tens and ones. • Die 0 to 9.** • Die 0 to 5.** • Shaker. • For each player, pencil and letter‑size paper. * Provided in the photomasters. ** Spinners may be used instead.

Rules of Stage (a) the game Preparation 1. The players fold their papers once each way to make 4 rectangular spaces.

These are used as bingo cards. 2. The first player throws the two dice. The 0 to 5 die gives the tens, the 0 to 9 die

gives the ones. 3. Suppose that he gets 2 tens and 4 ones. He takes the corresponding ten‑rods and ones from the box and puts these into the first space on his bingo card. 4. The other players do likewise in turn. 5. Steps 2, 3, 4 are repeated until each has filled all his spaces.

Num 2.10 Place-value notation (cont.)

92

The game 1. The pack of number cards is shuffled and put face downward on the table. 2. The players in turn turn the top card over, calling each two ways. E.g., “Three

tens, five ones; thirty‑five.” “Seven tens, zero ones; seventy.” “Zero tens,* four ones; four.”

3. It is important to speak each number in these two ways. After calling, the cards are put face down in another pile.

4. When a number is called corresponding to the ten‑rods and units in one of a player’s spaces, he takes these off and writes the numeral for them. 5. The first player to replace all his ten‑rods and units by numerals is the winner.

He calls “Bingo.” 6. The game continues until all the players have done likewise.

Notes * The question of whether zero may be omitted is considered later, in Num 2.12. (i) The game can be played with cards from 0 to 99, but this uses a lot of ten‑rods

and no new ideas are involved. (ii) The paper may be folded to make 6 spaces if desired.

Stage (b) This is played in the same way as Stage (a), except that instead of ten‑rods and units,

10‑cent and 1‑cent coins (genuine or plastic) are used.

Activity 1 introduces the final step into place‑value notation, in which an important part of the meaning of each digit (namely whether it means that number of ones, tens, hundreds . . . ) is not written down at all, but is implied by relative position. It is therefore very important that this meaning has been accurately and firmly established, which is the purpose of all the preparatory activities in earlier topics. Also, that the connection of the new notation with this meaning is established and maintained. Since the tens and ones no longer appear visibly, children now have to use their memory‑images instead. These images are exercised and consolidated by having the children speak aloud (e.g., “four tens, seven ones”) every time. Activity 2 relates the new notation to the individual meanings of each digit both as expressed in words (five tens, three ones) and as embodied in physical materials (ten‑rods and single cubes).

24

Discussion of activities

Num 2.10 Place-value notation (cont.)

93

Activity 3 uses all these connections:

All three modes of schema building, and all three modes of testing, are brought into use for this final step into place‑value notation.

Mode 1 building The new notation is related to physical embodiments.Mode 2 building In Activity 1, the new notation is communicated, both verbally

and in writing by a teacher. Activities 2 and 3 consolidate the shared meaning of this notation, by using it in a social context as basis for two games.

Mode 3 building The new notation is an extrapolation of the headed columns notation which they already know.

Mode 1 testing In Activity 1, children see column headings which confirm the accuracy of their readings in the ‘tens, ones’ form.

Mode 2 testing Activities 2 and 3 are games in which the players check the ac‑curacy of each other’s actions.

Mode 3 testing This is implicit in the transition from headed columns to place value, since if the new notation were not consistent with the one they already know, children would not accept it so readily as they do.

oBserVe aNd listeN reFlect discuss

written numerals (bothreading and writing them)

meanings ofindividual

digits

physical embodiments:ten‑rods, units

spokennumber‑words

Num 2.10 Place-value notation (cont.)

94

Num 2.11 caNoNical Form

Concept Canonical form as being one of a variety of ways in which a number may be written.

Abilities (i) To recognize whether a number is or is not written in canonical form. (ii) To rewrite a number into or out of canonical form.

The interchangeability of headed column notation and place‑value notation de‑pends on there being one digit only in each column, so that the first, second, third . . . columns reading from right to left always correspond one‑to‑one with the first, second, third . . . digits reading from right to left. However, it is one thing to note the advantages of having one digit only per column, and another to tell children that “We must not have more than one digit in any column,” or “We must not have numbers greater than nine in any column.” This is incorrect, because when add‑ing (e.g.) 57 and 85, we shall (temporarily) have numbers greater than nine in both columns; and when calculating (e.g.) 53 ‑ 16, children are told to take a ten from the tens to the ones column so that we can subtract 6. The present approach recognizes that there are a variety of correct ways of writ‑ing numbers, one of these being called canonical form. This has only one digit per column (which may be a zero), and has the advantage that in this case, and not otherwise, headed column notation can be replaced by place‑value notation. For this reason, canonical form is used unless there is a reason for using one of the alternatives – which we often need to do as a temporary measure. Here are some examples of the same numbers written in non‑canonical and canonical forms. For a given number there is only one canonical form, but many non‑canonical.

Tens Ones Tens Ones

Tens Ones Tens Ones

samenumber

samenumber

4 2 3 122 22

321

5 0 4 10

Canonical Non‑canonical

Discussion of concepts

95

Non‑canonical forms like the above often arise during calculations. To give the final answers in place‑value notation, facility in converting to canonical form is required. But place‑value notation depends on there being only one digit per column, so I think that headed column notation should be used whenever changes to and from canonical form are involved, until the children are absolutely clear what they are doing. The conventional notation, involving little figures written di‑agonally above, is a further condensation which may be quicker to write but is not consistent with this requirement. And now that calculators are readily available, it is understanding that is most important in written calculations. For speed, calcula‑tors are better. A further complication, usually ‘swept under the carpet,’ is that in non‑canonical form, place‑value notation makes an appearance within the headed columns. There is no mathematical inconsistency, but we are now using two notations in combina‑tion. I see no objection to this provided that we show both of them clearly. This is what I have tried to do in the activities which follow.

activity 1 Cashier giving fewest coins [Num 2.11/1] A game for up to five players. Its purpose is to present canonical form in a familiar

embodiment. Children should first have done Org 1.12 (in Volume 1) and Org 13.

Materials • Coins, 1‑cent, 10‑cent and, later, 100‑cent ($1). • Number cards 1 ‑ 49, or 1 ‑ 99 if preferred.* • Pieces of paper for cash slips. * Provided in the photomasters

What they do Stage (a) 1. One child acts as cashier, and has a supply of 1‑cent and 10‑cent coins. (She

should organize these to suit herself.) 2. The number pack is put face down on the table.

Num 2.11 Canonical form (cont.)

Tens Ones

Tens Ones

samenumber

samenumber

4 6

50

Hundreds

16

Tens Ones

Tens Ones

13 1614 6

3

9 15

Hundreds

HundredsHundreds

8

7

8

77

66 4 65

Canonical Non‑canonical

96

Num 2.11 Canonical form (cont.)

3. Each player in turn, turns over the top card and records on her own cash slip what she gets, using headed columns.

Say she gets 47. This entitles her to 47 single pennies, or their equal in value. She first writes 47 in the single cent column. 4. She hands it to the cashier. The cashier replies, “I want to use the fewest coins,” and returns the paper. 5. The player changes her request to 4 ten‑cent pieces and 7 single pennies, which the cashier accepts and pays. She records her agreement with a check mark, and the closing of the transaction with a line. 6. If a player turns a card showing less than 10, conversion is of course not re‑

quired. 7. After three rounds, each player counts her money, and if she has more than nine

coins of a kind, exchanges as appropriate with the cashier. 8. The winner is the one who has most, and she acts as cashier for the next round.

Extension A bonus of 100¢ ($1) may be earned by first adding the three underlined amounts,

converting, and predicting the result before checking physically. When the game is established, players may forestall the cashier’s refusal by mak‑

ing the conversion before passing her their cash slip. Both forms should however always be recorded: first the figure on the number card, then the equivalent value in canonical form. The whole object of this game is to establish that these are two ways of writing the same number.

Stage (b) (Return to this after Num 2.13, ‘Numerals beyond 100, written and spoken.’) It is played exactly as for Stage (a) except that two number packs of different colours

are used, one signifying dimes (10¢ coins) and the other pennies (1¢ coins). The conversion to canonical form may now require several steps. Suppose a player turns up 15 tens, 12 ones.

First we make separate conversions.

Then we combine their results.

Tens OnesHundreds¢ ¢ ¢

477√4

47Tens OnesHundreds

¢ ¢ ¢

12Tens OnesHundreds

¢ ¢ ¢

1521

1 5261

97

Num 2.11 Canonical form (cont.)

71

Tens OnesHundreds¢ ¢ ¢

46

174 6

1134

135

To begin with every step should be written, as illustrated. In this case it makes no difference whether the 12 singles or the 15 tens are converted first. With proficiency, however, the process of combining can be done mentally, and it is then easier to work from right to left.

A player who turns over (say) 46 tens, 71 singles will have further steps to take before conversion to canonical form is complete.

(One step has been done mentally here.)

activity 2 “how would you like it?” [Num 2.11/2]

This is similar to Activity 1, except that the cashier behaves differently. Instead of paying out the amount in the smallest number of coins, she acts like a bank cashier who asks, “How would you like it?” when we cash a cheque.

Suppose, as before, a player turns over 47. She records this, as before, but she might then ask for it to be paid like this

or this

or even like this.

98

Discussion of activities

4 71

Num 2.11 Canonical form (cont.) So the cashier needs to keep plenty of piles of ten single pennies. If the cashier runs

out of change before all have had three turns, she will have to ask the customers to help by giving her back some change in return for larger coins.

This game can be played at Stages (a) and (b), as in Activity 1. In both cases the final counting, and conversion to canonical form, will now be complicated, so Ac‑tivity 2 should not be tackled until Activity 1 is well mastered. At any time when a player has made a mistake, the conversions should be done with coins to check the pencil‑and‑paper conversions.

Children have already formed the concept of canonical form as it applies to the re‑peated grouping of physical materials, in Org 1.12 (SAIL Volume 1) and the topics leading up to it; and, for base 10, in Org 1.13. The present topic is concerned with the corresponding regroupings done mentally, and recorded using the mathematical notation shown in Activities 1 and 2.

This notation conveys, but more clearly, the same meaning as the makeshift devices which mostof us learned at school, suchas this: Canonical form is concerned with different notations for representing the same numbers. The symbolic manipulations practised here all represent regrouping exchanges such as those done in Org 1 with physical materials. Children now need to acquire facility at a purely symbolic level, so the base 10 physical material is replaced by coins. These provide an excellent intermediate material for the present stage, since they represent number values in a way which is partly physical, partly symbolic. This topic forms cross‑links with the various calculations which cause non‑ca‑nonical forms to arise, and practice with these will be found in the appropriate net‑works. But canonical form is a major concept in its own right, and other examples of it occur in later mathematics. It needs to be presented in a way which allows children to concentrate on learning this important concept by itself, without having at the same time to cope with other mathematical operations. oBserVe aNd listeN reFlect discuss

3

99

Discussion of concept

7 same number0

3 different number

0 different number3

Num 2.12 the eFFects oF Zero

Concept Zero as a place‑holder.

Ability In place‑value notation, to recognize when zero is necessary for giving their correct values to other digits, and when it is not.

In place‑value notation, it is the position first, or second, or third . . . in order from right to left which determines the value of a digit. So the numerals 04 and 4 both mean the same: 4 is in the ones place in each, and there are zero tens – explicitly in the first, implicitly in the second. However, the numerals 40 and 4 do not mean the same, since 4 is in the tens place in the first numeral and in the ones place in the second. In the numeral 40, zero acts as a place‑holder. By occupying the ones position, zero determines the meaning of 4. Though the zero in 04 is not necessary, it is not incorrect. Its use is becoming increasingly common, e.g., in digital watches, and in figures on cheques and else‑where for computer processing.

activity 1 “same number, or different?” [Num 2.12/1] A game for two or more players. They all need to see the cards the same way up. Its

purpose is for children to learn when the presence or absence of a zero changes the meaning of a numeral, and when it does not.

Materials • A double pack of single headed number cards from 0 to 9.* • A mixed pack of cards on which are the words ‘same number’ or ‘different

number,’ about 5 of each.* • An extra zero, of a different colour.* * Provided in photomasters

Rules of Stage (a) the game 1. Both packs of cards are shuffled and put face down on the table. 2. The player whose turn it is has the extra zero. 3. He turns over one card from each pile. Suppose he gets these. He now has to put down a zero so that what is on the table still means the same number as before. (It does NOT tell him to

“add a zero.” This would mean something quite different.) 4. If he does this correctly, the others award him a point.

5. The circulating zero now goes to the next player, who might turn over these cards.

7 same number

100

Num 2.12 The effects of zero (cont.)

7 same number0

3 different number

0 different number3

24

0 different number

2 different number0

4 2

4

different number

6. This would be his correct response.

7. A player who turns over 0 from the pile, together with ‘different number,’ still gets a point for saying, “It can’t be done,” and explaining why.

8. It is good for the players to speak aloud the numbers before and after putting down the zero. This makes it sound different if the numbers are different. E.g., in steps 3 and 4: “Seven ones.” “Zero tens, seven ones.” And in steps 5 and 6: “Three ones.” “Three tens, zero ones.”

Stage (b) (Return to this after Num 2.13, ‘Numerals beyond 100, written and spoken.’) This is

played as in Stage (a), except that they turn over two number cards from the pile and put them side by side.

Suppose a player gets this:

In this case (though not always) there are two correct responses.

If he gives both, and speaks both correctly, he is awarded 2 points.

activity 2 Less than, greater than [Num 2.12/2] Stage (a) A game for two players. Its purpose is to provide further examples of the effects of

zero.

Materials For each player: • A pack of single headed number cards 1‑9.* Shared between two: • A single zero card.* • An ‘is less than’ card.* • An ‘is greater than’ card, both as illustrated on the following page.* * Provided in the photomasters

What they do 1. The ‘is less than’ card is placed between the two players. Also on the table is the zero card.

2. Each player holds his own pack face down. Each takes the top card from his own pack and lays it on his side of the ‘is less than’ card.

3. The first player to have a turn then picks up the zero and must place it next to his own card to make a true statement.

101

Num 2.12 The effects of zero (cont.)

35is less than

<

75is less than

<

75is less than

<

75is less than

<

0

0

73is less than

< 25

Discussion of activities

Suppose that here it is the left‑hand player’s turn.

His correct response is: For this he gets one point.

Had it been the right‑hand player’s turn, one correct response would have been: (What is the other?)

In this case, only the right hand player can give a correct response.

If it was the left‑hand player’s turn, he could if he wished take a chance with an incorrect response. In that case, the other may say “Challenge,” and if the chal‑lenge is upheld the challenger gets two points.

When all cards have been used, the game is repeated using the ‘is greater than’ sign.

Stage (b) (Return to this after Num 2.13, ‘Numerals beyond 100, written and spoken.’) As with Activity 1, this can be played with larger numbers. Each player now begins

by putting down two cards. Several correct responses may then be possible, and players get a point for each which is both tabled and verbalized.

For example, in this situation the right‑hand player has three correct responses: 072, 702, and 720.

In this topic, children are now working at a purely symbolic level, without any sup‑port from physical materials. The activities involved are quite sophisticated, since they involve changing meanings for the same symbols. Agreement about these changing meanings depends on a shared schema for assigning values to symbols: this schema being (as we have noted) that of a condensed and sophisticated nota‑tion, in which much of the meaning is not explicitly written down at all, but is implied by relative positions. Any difficulties will usually be best dealt with by relating these activities to the more explicit headed columns notation; and, if necessary, by providing further backup in the form of base ten physical materials. These can be used to make very clear what a great difference in meaning can result from quite small changes in the positions of symbols.

oBserVe aNd listeN reFlect discuss

102

millions hundred-thousands

ten-thousands

thousands hundreds tens ones

hinged here

Num 2.13 Numerals BeYoNd 100, WritteN aNd sPoKeN

Concepts (i) That a particular digit can also represent a number of hundreds, thousands, ten‑thousands, hundred‑thousands, millions, according to whether it comes third, fourth, fifth, sixth, seventh in order, reading from right to left.

(ii) That each new position in this order (together with the associated word) repre‑sents groups of ten of the group just before.

Abilities (i) To identify separately which digits represent hundreds, thousands, ten‑ thou‑sands, hundred‑thousands, millions.

(ii) To write numbers in expanded notation. (iii) To read aloud numerals having up to seven digits. (iv) To match these with physical representations.

Here we are extrapolating the place‑value concept from its limited application to tens and ones successively to hundreds, thousands, . . . . The essence of this is a repetition, each time we move one place further from right to left, of the grouping process, each new larger group containing ten of the group just before. This extrapolation can and should be begun physically. Since it is not likely that a million units are available, we are led naturally to an intermediate representation which is partly physical, partly symbolic.

activity 1 big numbers [Num 2.13/1]

This is a teacher‑led activity for as many as can see the chart properly. Its purpose is to extrapolate children’s use of place‑value notation up to the seventh place, repre‑senting millions.

Materials • A chart prepared as in the illustration below. • Base ten material. • A skeleton metre cube (representing a million) would be very helpful if possible.

This would have to be filled with units to represent a million physically.

Discussion of concepts

103

millions hundred-thousands

ten-thousands

thousands hundreds tens ones

Num 2.13 Numerals beyond 100, written and spoken (cont.)

Stage (a) 1. Place the chart where the children can see it with the hinged section folded under

so that only the right‑hand section is visible (hundreds, tens, ones). 2. Point to the ones section, and (with the base‑ten material on view) ask, “Which

do we put here?” 3. The answer is “units” [or “ones”], and you put a specimen unit in the space, say‑

ing “Like these.” 4. Similarly for the tens and hundreds sections, the specimens in these cases being

a ten‑rod and a ten‑square respectively. They will already know that the number of units in a ten‑square is called a hundred.

Now go straight on to Stage (a) of Activity 2, ‘Naming big numbers.’ Stage (b) 5. Now open the chart out to show all the sections. Put a specimen from the base‑

ten material in the ones, tens, hundreds.

6. Continue, pointing to the thousands space, “What do we put here?” The answer is “A ten‑cube” and you put a specimen in the thousands space. The number of units in a ten‑cube is called a thousand.

7. For the next space ten‑thousands, a problem arises. Even if there are ten ten‑cubes available, they would make a tall and unstable column. Ask the children whether, instead, they can imagine a ten‑rod made up of thousands in this space.

8. Next we come to the hundred‑thousand space. Ask “Now, what should we imagine in this space?” (pointing). The answer is a ten square of thousands; and for the next (millions) space, a ten‑cube of thousands.

9. It would be helpful to support their imaginations with a skeleton metre cube (representing a million).

Big numbers [Num 2.13/1]

104

Num 2.13 Numerals beyond 100, written and spoken (cont.)

hundreds tens ones

3 7

7

hundreds tens ones

3 0

activity 2 naming big numbers [Num 2.13/2]

For as many players as can see the material the same way up. (Otherwise the order gets reversed.)

Materials • A chart as used in Activity 1. • A box of base ten material. • A pack of single headed number cards 0 to 9.* • Pencil and paper for writing numbers in expanded notation. * Provided in the photomasters

What they do Stage (a) 1. The pack is shuffled and put face down on the table. 2. The first player picks up the top card and puts it face upward beside the chart, and says (in this example) “Three ones: three,” and puts the three units into the correct space.

3. The next player picks up the next card, and puts it on the right of the one already there. He now says (in this example) “Three tens, seven ones: thirty‑seven,” puts down material accordingly, and writes the number in expand‑

ed notation and in place value notation: ‘30 + 7 = 37.’

4. The next player picks up the next card, and again puts it on the right of the one already there. She now says (in this example) “Three hundreds, seven tens, zero ones: three hundred seventy,” puts down material accordingly, and writes:

‘300 + 70 + 0 = 370.’ 5. The chart is now cleared of material, and the cards returned randomly to the

pack. 6. Steps 2 to 4 are repeated until the children have mastered this stage. 7. Now please read the directions at the end of Stage (b), on the next page.

hundreds tens ones

3

hundreds tens ones

3 7

7

hundreds tens ones

3 0

105

Num 2.13 Numerals beyond 100, written and spoken (cont.)

5

5

5

175

7 1 6 2

7

Stage (b) Materials • As in Stage (a) to begin with. • Pencil and paper for writing expanded notation and for scoring.

What they do This is played a little differently. The base ten material is no longer used. For the first few rounds the chart from ones to millions is left on display as a reminder, after which it is put away and the game continued mentally.

1. The 0 ‑ 9 pack is put face down on the table. 2. The players in turn pick up the top card and put it down on the right of those

already there, as in stage (a).

3. The first player says “Five ones: five.”

4. The next player says “Five tens, seven ones: fifty‑seven,” and writes ‘50 + 7 = 57.’

5. The next player says “Five hundreds, seven tens, one one: five hundred, seventy‑one”

and writes ‘500 + 70 + 1 = 571.’

and so on.

6. This player should say “Five ten‑thousands, seven thousands, one hundred, six tens, two ones: fifty‑seven thousand, one hundred,

sixty two,” writing ‘50 000 + 7 000 + 100 + 60 + 2 = 57 162.’ 7. Scoring for a correct answer is 1 point for each correct digit and 1 point for cor‑

rectly expressing the number in expanded form. 8. If no one makes a mistake, the round continues to the millions, and the player to

whom this falls will score eight points. 9. In this case, the following round starts with the next player. This ensures (pro‑

vided that the number of players is not seven) that the millions turn will come to a different player each time.

As soon as children are confident with Activity 1, ‘Big numbers,’ Stage (a), they should continue straight on to Stage (a) of Activity 2, ‘Naming big numbers.’

After that, there are two paths open to them. (i) They may return to Num 2.11 and Num 2.12 doing these activities at the more

difficult stages involving hundreds, tens, ones. (ii) They may continue to Stages (b) of Activities 1 and 2 (‘Big numbers,’ and

‘Naming big numbers’), and then follow path (i). These two paths may also be followed alternately.

106

Num 2.13 Numerals beyond 100, written and spoken (cont.)

This topic makes considerable use of Mode 3 schema building: extrapolation of an existing pattern, and combining ideas which they already have. The pattern is that of repeatedly putting together the same number of groups to make the next group larger. This pattern was built up using physical materials (Mode 1) in Org 1, topics 10 to 12 in SAIL Volume 1 and topic 13 in this volume. To start with it was based on numbers small enough to be subitized (perceived without counting). Then it was extended to base 10, using prefabricated physical materials so that the process of putting together physical materials to form larger groups was replaced by its mental equivalent. In Activity 1, Stage (a), this pattern is picked up again, using physical materials by way of review. In Stage (b), it is continued much farther. For this, we let go of the physical materials and we use our imagination. The children will in these ac‑tivities be combining this extended pattern with words they have probably already heard: “hundred,” “thousand,” possibly even “million.” Note how in Activity 2, every digit changes its place value at every turn, thereby emphasizing how the meaning depends both on the digit itself and on its location. Expanded notation is introduced explicitly at this point, at first with the support of base ten material and oral interpretation. You may wish to postpone the introduc‑tion of expanded notation for some students. Others may be ready to make use of exponential notation, as in:

57 162 = 5 x 104 + 7 x 103 + 1 x 102 + 6 x 10 + 2

oBserVe aNd listeN reFlect discuss

Discussion of activities

Naming big numbers [Num 2.13/2]

107

Num 2.14 rouNdiNg (Whole NumBers)

Concept That of rounding a given whole number to the nearest ten, hundred, thousand, . . . .

Ability To round any given number as above.

Sometimes too much information is as unhelpful as too little. For example, if we want to compare the populations of several large cities, this can be grasped more easily if they are not shown to the last person. To the nearest hundred would be accurate enough, or even perhaps the nearest thousand. Rounding is thus a method of giving a suitable amount of information for a particular purpose.

activity 1 run for shelter [Num 2.14/1]

A game for up to four players. Its purpose is to introduce the concept of rounding in the context of a number line. (In this case, it is a number line with embellishments!)

Materials • Game board (see Figure 3).* • Die, marked 1, 2, 4, 5, R, R. • Shaker (or the die may be rolled from the hands.) • Markers, one for each player.† * Provided in the photomasters † These need to be quite small. I use short pieces of

coloured milk straws with a small blob of Blu‑tak (or Plasticine). [See illustration.]

What they do 1. The board represents a winding track with a shelter at every tenth location. The game is of the familiar kind, in which players roll the die and move forward that number of places. More than one player may be at the same location.

2. There is, however, this difference: if the die shows R, this means Rain. It also means Run to the nearest shelter.

3. Which shelter this is depends on the present location of the player. If she is at 1, 2, 3, 4 after a ten, she has to return to the ten just passed. If she is at 6, 7, 8, 9 after a ten, she runs forward to the next shelter ahead. If she is on a 5, she is half way between two shelters, so naturally chooses to run to the one ahead.

4. The winner is the first to reach the camp at end of the track. For this either the exact number, or an R, is required. The others may continue if they wish.

5. Explain that what they have been learning is called rounding to the nearest ten. Now the R also means rounding.

Note If this game is found too long, it may be played finishing at (say) 50 or 60.

Discussion of concept

108

figure 3 ‘run for shelter’ game board

Num 2.14 Rounding (whole numbers) (cont.)

109

activity 2 rounding to the nearest hundred or thousand [Num 2.14/2]

An activity for as many children as can see the materials the right way up. (Oth‑erwise the direction gets reversed.) Its purpose is to make explicit and extend the concept of rounding, and to help the ability become independent of the number track.

Materials For the group: • A card with number lines, as illustrated in Figure 4.* • A pack of single headed number cards, 0 to 9. * • Something to point with which does not make a mark. • A plain card on which to lay out the number cards is also useful. * Provided in the photomasters

Stage (a) Teacher-led introduction 1. Show the number line card to the children, and explain that this is part of a long

number line: too long to show the whole of it. Ask them to identify the un‑marked points.

2. Put out three number cards to show a number between 200 and 300: E.g.:

3. Ask one of them to point to its ap‑proximate position on the number line. In this case, this is somewhere between the points for 270 and 280.

4. “So if we go to the nearest hundred, where would this be?” Ask them to say the number as well as pointing. (300.) “This is called ‘Rounding to the nearest hundred.’”

5. Ask them to put into words what decides which way they go. Something like “If it’s less than half way, you go back. If it’s at or above half way, you go on.”

6. Put out other numbers between 200 and 300, some nearer 200 and some nearer 300. And ask the children in turn first to point to it on the number line, and then say the result of rounding to the nearest hundred.

7. Now lay out another number which does not begin with 2, e.g.:

8. “For this we need another part of the number line.” (Pointing to the next number line down, and indi‑cating the ‘hundreds’ markers:)

“What would these two points have to be? (600 and 700.) So where would this be, approximately? (Between 610 and 620).

9. Can you round this to the nearest hundred? (600.) 10. Repeat as long as seems necessary.

2 7 3

6 1 4

Num 2.14 Rounding (whole numbers) (cont.)

Suggested sequence for the introduction

110

200 300

7000 8000

figure 4 number lines card for ‘rounding to the nearest 100 or 1000’

Num 2.14 Rounding (whole numbers) (cont.)

111

Rules of the game

3 7 4

3 7

Num 2.14 Rounding (whole numbers) (cont.)

Stage (b) (for them to continue on their own) 1. The pack is shuffled and put face down. 2. The child whose turn it is to begin turns over the top three cards and puts them

in a row, as in Stage (a). 3. All write down what this is when rounded to the nearest hundred. 4. The child whose turn it is reads out her result. Others say whether they agree or

not. Any disagreements are resolved as described in step 6. 5. The next child puts the top card from the pack face up on the right of those already

there, and removes the left hand card which she puts in a separate face down pack. 6. Steps 3 and 4 are then repeated. Each time, the child whose turn it is reads out

her result. If there is disagreement, it is for her to explain, with the aid of the number line, why she thinks that her answer is the right one.

7. This continues until the children are fluent at rounding to the nearest hundred. When the pack is used up, it is shuffled and replaced for a fresh start.

Stage (c) As for Stage (b), but using four cards, and rounding to the nearest thousand. Thou‑sands number lines are available, if required, on the same card as the hundreds number line already used.

activity 3 rounding big numbers [Num 2.14/3]

A game for as many children as can see the materials the right way up. It follows Activity 2 closely, and its purpose is to extend the concept and ability of rounding numbers up to the nearest million, and beyond if they choose. It requires good con‑centration, and offers a challenge to the more able children.

Materials • Single headed number cards, as for Activity 2. For this game, a double pack may be preferred.

• Pencil and paper for scoring. 1. One player is appointed score‑keeper. The pack is shuffled and put face down

on the table. 2. This game is similar to Stage (b) of the previous activity, except that two cards

are turned up by the first player and no card is removed by subsequent players. The number of cards showing thus increases by one for each successive player.

3. The first player picks up the top two cards, puts them face up side by side, and says (in this example) “Thirty‑seven, rounded to the nearest ten, is forty.” For this she scores 1 point.

4. The next player picks up the top card, puts it face up on the right of those already on the table, and says (in this example) “Three hundred seventy‑four, rounded to the near‑est ten, is three hundred seventy. Rounded to the nearest hundred, is four hundred.” Score, 2 points.

112

5. The next player picks up the top card, puts it face up on the right of those already on the table, and says (in this example) “Three thousand seven hundred forty‑five, round‑ed to the nearest ten, is three thousand seven hundred fifty. Rounded to the near‑est hundred, it is three thousand seven hun‑dred.” (Not three thousand eight hundred, a mistake which is sometimes made. A number line will make this clear if neces‑sary.) Rounded to the nearest thousand, it is four thousand.” Score, 3 points.

6. If no one makes a mistake, the game continues to the millions, i.e., with seven cards face up, and the player to whom this falls will score 6 points.

7 The next player shuffles the pack and starts again, with two cards as in step 3. They then continue with steps 4 to 7 as before.

8. If a player makes a mistake, she scores the points she has earned, but the turn then passes to the next player.

9. This game requires much concentration. Therefore, if anyone makes any kind of interruption, except “Yes” or “No,” while a player is having her turn, this player is allowed a penalty‑free mistake.

Once again the number line is used as a powerful support to our thinking. Activity 1 embodies it in a game which introduces rounding in its simplest form. Activity 2 extrapolates the concept to include rounding to the nearest hundred or thousand. Activity 3 requires mental concentration, which needs peace and quiet: and it was during field testing that I proposed the rule in step 9 of the last activity. All five children unanimously agreed that it was a good rule, although most of them had been interrupting! So in this activity, the children are learning more than the math‑ematics itself.

oBserVe aNd listeN reFlect discuss

Discussion of activities

7 4 5 3

Num 2.14 Rounding (whole numbers) (cont.)

113

[Num 3] Addition A mathematical operation which corresponds with a variety

of physical actions and events

Num 3.9 AddiNg, results up to 99

Concept Adding when the results are between 20 and 99.

Ability To add 2-digit numbers, the results still being 2-digit numbers.

What is new here is not the concept of addition, but extension of the ability to do this with larger numbers.

Activity 1 Start, Action, Result up to 99 [Num 3.9/1] This is a continuation of Num 3.7/1 (in Volume 1), ‘Start, Action, Result over 10,’

with larger numbers. Materials • SAR board as illustrated below in step 1.* • Start and Action cards with assorted numbers from 1 to 49.* • Base ten material, tens and ones. • Paper and pencil for each child. * Provided in the photomasters

What they do Stage (a) Headed columns and materials (The steps for this are the same as in Num 3.7/1, and are repeated here for conven-

ience.)

What they do 1. TheStartandActioncardsareshuffledandputinpositionasusual,andthetoptwo cards turned over.

discussion of concept

Start

Action

S

A

Do and sayTens Ones

Result

46

+38

114

2. Tens and single cubes are put down as shown on these cards by one of the chil-dren, who describes what she is doing and what it corresponds to on the SAR board. E.g., “The Start card means ‘Put down four tens and six ones.’” “The Action card means ‘Put three tens and eight ones more.’” Each child records thisonherownpaper,whichshefirstrulesintoheadedcolumns.Theboard,andrecords, will now appear as below.

3. The next stage is shown below.

4. Finally, if there are (as in this case) more than ten ones, ten of them are ex-changed for a ten-rod. This is transferred to the tens column. The children again record this individually (as illustrated on the following page).

Num 3.9 Adding, results up to 99 (cont.)

Start

Action

S

A

Do and sayTens Ones

Result

46

+38

Tens Ones

6 8

4+ 3

Start

Action

S

A

Do and sayTens Ones

Result

46

+38

Tens Ones

6 8

14

4+ 3

7

115

5. Childrencomparetheirfinalresults.Thereshouldbenodifference,butifthereis, they will need to repeat the process and check each step together.

6. The board is cleared, and steps 1 to 5 are repeated with different numbers.

Stage (b) Headed columns, no materials 1. The SAR board and base ten materials are no longer used. 2. Start and Action cards are turned over as before, and all the stages of addition

are written in headed column notation as in stage (a).

Stage (c) Place-value notation 1. Start and Action cards are turned as before. If the addition would result in 10

or more in the units position, a streamlined form of headed column notation is used.

2. Here is a simple example. First the addition 3 7 to be done + 2 5 is written.

3. Adding 7 and 5 T Ones gives a result 3 7 in 2 digits, so + 2 5 we need headed 5 12 columns.

4. Regroup. T Ones 3 7 + 2 5 5 12 6 2

Start

Action

S

A

Do and sayTens Ones

Result

46

+38

Tens Ones

6 8

148 4

4 + 3

7

Num 3.9 Adding, results up to 99 (cont.)

116

Num 3.9 Adding, results up to 99 (cont.) 5. Finally, write the 3 7 result in place value + 2 5 notation. 6 2

6. The whole may be set out like this. T Ones 3 7 3 7 +2 5 + 2 5 6 2 5 12 6 2

This result is written last.

Activity 2 odd sums for odd jobs [Num 3.9/2] A game for about four children. Its purpose is to use the skills learned in Activity 1.

Materials • 1 die bearing only 1s and 2s [for tens]. • 1 normal 1-6 die (or 0-9, if you have one) [for ones]. • 1 set of ‘odd job’ cards.* • 1 set of ‘target’ cards.** • Plastic or real money: 10¢, 5¢, 1¢ coins. • Money boxes with a slit in the lid for all but one of the players. • Paper and pencil for each player. *Written on these are jobs they might do such as “Wash car,” “Clean all downstairs

windows.” (provided in the photomasters) **Written on these are the kind of things they might be working for, such as “Swim-

ming, 60¢ for two,” “Present for sister, 78¢.” (provided in the photomasters)

What they do 1. One child acts as ‘Mom’ or ‘Dad.’ 2. Each child picks a target from the target cards which are held in a fan or spread

out face downwards on the table. 3. Then,startingwiththechildontheleftof‘Mom’or‘Dad,’thefirst‘doesajob’

byturningoveracard,andsincethereisnofixedrate,throwsthedicetoseewhat she is paid.

4. The player acting as ‘Mom’ gives her the money in coins. 5. She puts these in her money box, which she may not open without permission. 6. Each takes her turn, putting the money in her box each time. 7. To know when she has enough money, each child keeps a record, adding on her

earnings each time. 8. Thefirstwhothinksshehasenoughaskspermissiontoopenhermoneybox,and

her recorded total is checked by her ‘parent’ against the money in the box. 9. If correct she gets to be ‘Mom’ next time. 10. If not she has to go on doing jobs!

117

Activity 3 Renovating a house [Num 3.9/3] A co-operative game for 4 children. Its purpose is to practise the skills developed in

Activities 1 and 2 in another play situation.

Materials • Gameboard, see Figure 5.* • On cards:* house chimney windows doors • Tens die, numbered 1 & 2 only. • Ones die, numbered 0 to 9. • Play money: $10 bills, $5 bills, and $1 coins. • Slips of paper, pencils. * Provided in the photomasters It is a good idea to introduce the game with costs in round numbers, e.g., window

$30, door $50, and progress to harder numbers as shown. For this, two houses will be needed or adhesive labels used, after laminating, on which there are various sets of prices.

window$27

window$27

window$32

window$32

door frame $34

door$55

chimney$63

sidedoor$46

window$28

Figure 5 Renovating a house

Num 3.9 Adding, results up to 99 (cont.)

118

What they do 1. One child acts as banker, one as building supplies dealer, and two as a young couple who are saving what they can each week and putting the money towards parts for their house. This is an old house which they have bought cheaply and are renovating.

2. Each of the couple throws the dice to determine how much they have saved from their earnings that month.

3. They record these amounts on a slip of paper, and add them together.

4. They take their slip to the banker, who checks their total and gives them cash in exchange, keeping the slip.

5. When they have enough, they go to the building supplies dealer and buy a door, window, or chimney. The banker may be asked to exchange larger bills for smaller.

6. When the house is built, they may play another game, exchanging roles.

Activity 4 Planning our purchases [Num 3.9/4] For a small group. This activity uses the same mathematics for a situation which

is the reverse of Activity 2. Instead of accumulating money for a predetermined single target (which may be exceeded), they have a given amount of money which they plan to spend on a variety of purchases, keeping to a total cost within the given amount.

Materials • A variety of labelled articles for the shop.* • Slips of paper or card.** • Plastic money. • Pieces of paper and pencils for each child. *Varied prices in the range of 48¢, 19¢, 7¢ and a few penny objects. **On these are varying sums from 50¢ to 90¢, in tens.

What they do 1. Onechildfirstactsasbanker,andhaschargeofthemoney;thenasshopkeeper. 2. The other players draw slips of paper to discover how much they have to spend. 3. Each child takes her slip to the banker to get cash. 4. She then makes a shopping list which she totals. 5. When ready, she goes to the shop with her money and shopping list. 6. The shopkeeper then sells her the goods, naturally making sure that she receives

the correct payment. Shoppers can use penny objects to make up an exact amount, thereby avoiding the

necessity for giving change. Children could also ask the banker to change a 10¢ coin into 5¢ and/or single pennies, which is good practice for mentally changing between tens and singles. Receiving change is however something the children will already have experienced outside school, so you may prefer to leave out the penny objects and let the shopkeeper give change instead.

Num 3.9 Adding, results up to 99 (cont.)

24 + 17 41

119

Activity 5 Air freight [Num 3.9/5] Amoredifficultadditionactivity,forchildrenplayinginpairs.

Materials • Cards with pictures of objects together with their weights.* • Cards representing containers. * Suitable pictures might be bags, suitcases, small items of furniture, domestic items

like tape recorders, televisions, etc. It is good to have several assorted sets of these. For example, a set having 19 cards,

each with a picture from a mail order catalogue, marked with the following weights:

This makes a total of 599 kg, which it is possible to pack into 6 containers (5 holding 100 kg each, and 1 holding 99 kg). You could use other combinations of weights. It is probably easiest to choose them by breaking down 100 kg’s initially,

Num 3.9 Adding, results up to 99 (cont.)

Air freight [Num 3.9/5]

64 kg, 54 kg, 52 kg, 48 kg, 46 kg, 42 kg, 39 kg, 38 kg, 35 kg, 31 kg, 28 kg, 22 kg, 20 kg, 19 kg, 17 kg, 15 kg, 12 kg, 9 kg, 8 kg

e.g., 62, 27, 11. 53, 17, 22, 8, 49, 33, 18, 35, 31, 19, 10, 5 etc.

120

What they do 1. The object is to pack all the objects into the smallest possible number of contain-ers. No single container may weigh more than 100 kilograms.

2. They may work however they choose. One way is for one to act as packer, and the other as checker who makes sure that no container exceeds 100 kg.

3. If there are several pairs doing this activity at the same time, they may compare results to see which pair is the best at packing the containers.

These involve the extrapolation of techniques already learned to larger numbers, and the combining of concepts already formed. So the schema building involved in this topic is a good example of Mode 3: creativity. Grouping and regrouping in tens from Org 1, adding past ten from the topic just before this, and notations for tens and singles from Num 2, are the chief concepts to be synthesized. In Activity 1, stage (a) renews the connection between written addition and its meaning as embodied in base-ten materials. Stage (b) is transitional to addition using place-value notation in stage (c). In all these stages, place-value notation sometimes occurs in combination with headed columns. This hybrid was only adopted after careful analysis and discussion with teachers. It is implicit in the conventionalnotation,whensmall‘carrying’figuresareused;soisitnotbettertoshow clearly what we are doing, especially since it represents such an important part of the calculation? The layout suggested is very little slower than the tradi-tional one – and if speed is the main object, then calculators are the best means to achieve it. Activity 2 has been devised in the form shown to introduce a predictive element – checking the total on paper against coins in the money box. This you will recog-nize as an example of Mode 1 testing. Activities 2, 4 and 5 introduce multiple addends. In Activity 2, this is done by stages, each total being recorded before the next one is added. This is how we add mentally, so it is a good preparation for Activity 4 where the running totals need notberecordedunlesschildrenfindithelpful.Activity3,andthemoredifficultActivity 5, have been included to give a further choice of activities in this sec-tion, since it is good for children to get plenty of assorted practice at addition with regrouping.

oBserVe ANd listeN reFleCt disCuss

Num 3.9 Adding, results up to 99 (cont.)

discussion of activities

121

Num 3.10 AddiNg, results BeyoNd 100

Concept Adding when the results are from 100 to 999.

Ability To add 2 or 3 digit numbers, results in the above range.

As in Num 3.9, what we are concerned with in this topic is the further extension of their existing abilities in addition.

Activity 1 Start, Action, Result beyond 100 [Num 3.10/1]

This repeats Activity 1 of the last topic with hundreds, tens and ones. Its familiar-ityshouldgivechildrenconfidenceintacklingtheselargernumbersbeforetheyusethem in the new activities which follow.

Material • SAR board as for Num 3.9 but with an extra column for hundreds.* • Start and Action cards with assorted numbers between 1 and 499.* • Base ten material, hundreds, tens and units. • Pencil and paper for each child. * Provided in the photomasters

Stages (a), (b), (c) These are the same as for Num 3.9/1 except for the larger numbers.

Activity 2 Cycle camping [Num 3.10/2]

A board game for 2 or more. Its purpose is to exercise children’s skills in addition, relating this to what they have learned on the number line.

Materials • Board (see photomaster in Volume 2a). • Markers in the form of tents. • Die marked 1 - 6. • Die marked 0 - 9. • Shaker.

What they do 1. The board represents a long and winding road through beautiful countryside. 2. The players begin at 0, and in turn throw the dice to determine the distance to be

travelled for the day. 3. The 1 - 6 die gives the tens, and the 0 - 9 die gives the ones. The result is a vari-

able daily distance from 10 to 69 km. Children may use their imaginations to explain these.

4. Adding is done by ‘counting on’ separately in tens and in ones, a decade at a time for the tens.

5. Before moving, a cyclist points to where he thinks the day’s kilometres will take him. 6. The others check and say ‘Agree’ or ‘Disagree’. If the cyclist’s prediction is

incorrect, he has a puncture that day and cannot move. 7. Tofinish,inthisgametheexactnumberneednotbethrown.Anynumberof

kilometres equal to or greater than the remaining distance will do.

discussion of concept

122

Activity 3 one tonne van drivers [Num 3.10/3]

A board game for up to four players. Their roles as van drivers give them plenty of practice in addition of numbers with sums up to one thousand.

Materials • Game board, see Figure 6.* • Cards representing vans, one for each player.* • Cards representing loads.** • A calculator. • Pencil and paper for each player. • Paper clips, one for each player. * Provided in the photomasters

** Thechoiceofloadsaffectsboththelengthandthedifficultyofthegame. As a start I suggest you try the following.

6 large: 509, 576, 624, 697, 718, 745. 8 medium: 132, 194, 246, 287, 355, 369, 424, 431. 12 small: 12, 17, 21, 25, 34, 39, 40, 46, 65, 78, 89, 96.

What they do 1. One player acts as supervisor, the others are van drivers. 2. The supervisor has a calculator, the drivers do not. 3. The load cards are piled face down at the depots, which have respectively large,

medium, and small packages. 4. The drivers have cards representing vans of 1 tonne (1000 kg) capacity. They

drive these to the depots, and bring back a load to the warehouse. The load cards are kept on the vans by paper clips.

5. Players start at the warehouse, and move in turn. In a single move they may do any one of the following.

(a) Drive to any depot and take what is offered (the top card). (b) Stay where they are and take another item. (c) Return to the warehouse. (d) Unload at the warehouse and have the weight of their load checked by the

supervisor. 6. If, when his load is checked, it is found to be overweight (more than 1 tonne),

the driver must wait for 2 turns at the warehouse while his van is overhauled and tested. The excess weight is not credited.

7. The drivers use pencil and paper to add up their loads as they go along. If of-fered a package which would make them overweight, they should refuse it. This uses up their turn. The package is replaced at the bottom of the pile.

8. The game ends when all the goods have been collected and unloaded at the warehouse. The winner is the driver who has brought back most. He becomes the supervisor for the next game.

Num 3.10 Adding, results beyond 100 (cont.)

123

Figure 6 One tonne van drivers

Num 3.10 Adding, results beyond 100 (cont.)

124

Num 3.10 Adding, results beyond 100 (cont.)

Activity 4 Catalogue shopping* [Num 3.10/4]

An activity for a small group. This gives further uses for their addition skills. Since 1 dollar = 100 cents, working in dollars and cents is equivalent to working in hun-dreds, tens and units.

Materials • Pages from a mail order catalogue, pasted on card and headed, e.g., ‘$100 to spend.’ or ‘$25 to spend.’ • Pencil and paper for each child.

What they do 1. The children are given a catalogue card along with pencil and some pieces of paper.

2. They are asked to prepare a list of items to be ordered, within the total amount statedatthetopofthecard.Theyshouldfirstmakearoughlistandthenwriteout a tidy order.

3. Initially, headed column notation should be used to make the correspond-ence obvious.

4. Later this may be written as . . . using place-value notation. 5. At the present stage we are still using

cents as our units, and a dollar is just another name for 100 of these. Later, when the dollar is taken as the unit, the ‘period’ becomes a ‘decimal point’ and the next two digits represent tenths and

hundredths. Decimal fractions are still in the future, but this transitional notation is excellent preparation.

* For this activity I am indebted to Mrs. A. Cole, Headteacher of Loveston County Primary School, Dyfed.

We begin with the familiar ‘Start, Action, Result’ in the same three stages as be-fore. This picks up the group of related concepts which are by now well estab-lished, and which we want them to extrapolate further. ‘Cycle camping’ (Activity 2) uses a number line, in the form of a game, as another way of adding. Though the results go past 100, the new numbers to be added on do not. With this ground-ing in the expanded technique, children should be ready for Activities 3, ‘One tonnevandrivers,’and4,‘Catalogueshopping.’Thefirstoftheseisalittlelike‘Air Freight’ from the previous topic, but it also involves comparing possible out-comes of the moves they might make. The other is a budgeting activity. In adult life, planning the use of money and other resources – time, labour – is one of the major uses of arithmetic. This is one of the reasons I have introduced games of this kind.

oBserVe ANd listeN reFleCt disCuss

527¢

$5.27

dollars dimes cents 100¢ 10¢ 1¢

5 2 7

discussion of activities

125

[Num 4] Subtraction taking away, comparison, complement, Giving change

Num 4.6 GiviNG chaNGe

Concept Paying a required amount by giving more and getting change.

Abilities (i) To give the correct change. (ii) To check that one received the correct change.

This is another contributor to the comparison aspect of subtraction. In this case, the larger number is the amount tendered, the smaller number is the cost of the purchase, and the difference is the change.

activity 1 change by exchange [Num 4.6/1] An activity for 3 or 4 children (not more). Its purpose is to ‘spell out’ with the coins

themselves what is happening when we give or receive change.

Materials • Play money: 30¢ for each customer made up as in step 2, following; 1¢ and 5¢ coins for shopkeeper.

• A ‘till’ (tray with partitions). • Pictures on cards representing objects for sale, with prices marked, all less than

10¢.

What they do 1. One child acts as shopkeeper, the rest as customers. 2. Thecustomersstartwith30¢,madeupoftwo10¢coins,one5¢coin,andfive

1¢ coins. The shopkeeper has a plentiful supply of 1¢ and 5¢ coins. 3. The shopkeeper sets out her wares. If there is not enough table space for all the

goods, some may be kept ‘in the stock room’ and put out later. 4. The customers in turn make their purchases one at a time. 5. To start with, they pay with exact money. When they no longer have the exact

money for their purchases, they pay by giving more and getting change. 6. Suppose that a customer asks for a 6¢ apple, and hands a 10¢ coin to the shop-

keeper. 7. The shopkeeper says “I have to take 6¢ out of this, so I need to exchange it.”

She puts the 10¢ coin into her till and takes out 10 pennies. (With experience, a combination of 5¢ and 1¢ coins will be used.)

8. Spreading these smaller coins out, she then says “I’m taking 6¢ for your apple” and does so. “The rest is your change: 4 cents.”

9. The shopkeeper gives the apple and the 4¢ change to the customer, who checks that she has received the right change.

Discussion of concepts

126

activity 2 change by counting on [Num 4.6/2] A continuation from Activity 1, for 3 to 6 children. (May be included or by-passed,

at your discretion.) It is placed between Activities 1 and 3 to relate to its mathemati-cal meaning the method of giving change often used by cashiers.

Materials • The same as for Activity 1.

What they do 1 - 4. are the same as in Activity 1. 5. The method of giving change is now different. Assume as before that the cus-

tomer has handed a 10¢ coin to the cashier for a 6¢ apple. The cashier goes to the till and picks up coins to make the total up to 10¢, saying to herself “7, 8, 9, 10.”

6. She says to the customer “6¢ for your apple,” and then, while counting the change into the customer’s hand, “7, 8, 9, 10.”

Note that with this method, the amount of the change is not explicitly stated or writ-ten.

activity 3 till receipts [Num 4.6/3] A continuation from Activity 2, for 3 to 6 children. Its purpose is to relate the kind of

subtraction involved in giving change (comparison) to the conventional notation for subtraction.

Materials • The same as for Activities 1 and 2, and also • A pad of till receipts. (See illustration below.)

What they do 1 - 4. are as in Activity 1. 5. Having arrived at the right change by any means she likes, the shopkeeper then writes a till receipt for the customer. 6. She shows it to the customer like this, before removing it from the pad and handing it to the customer.

Num 4.6 Giving change (cont.)

127

7. This is what the customer receives, together with her purchase and change.

Once again, there is more here than meets the eye. The counting on method for giving change, as usually practised in shops, produces the correct change and al-lows the customer to check. But it does not say in advance what amount this will be, nor does it lead to subtraction on paper. So in Activities 1, 2, 3, we have a sequence. In Activity 1, the emphasis is on the concept itself of giving change, using the simplest possible way of arriving at the amount. Note that the customers begin with assorted coins, so that the activity does not begin with giving change, but with paying in the direct way. Giving more and receiving change is then seen as another way of paying the correct amount. We found that when this approach was not used, some children continued to give change even when the customer, having collected the right coins by receiving change, had then paid the exact amount! This shows how easily habit learning can creep in instead of understanding, and also how important it is to get the details rightintheseactivities.Activity2usescountingonasamethodforfirstproducingthe correct change, and then allowing the customer to check. Finally, Activity 3 transfers this to paper and makes explicit the amount of change which the customer should receive. It also relates this new aspect of subtraction to the notation with which children are already familiar. This helps to relate it to the overall concept of subtraction.

OBSeRve aND LiSTeN ReFLecT DiScUSS

Num 4.6 Giving change (cont.)

Discussion of activities

128

Num 4.7 SUBTRacTiON wiTh aLL iTS meaNiNGS

Concept Subtraction as a single mathematical operation with 4 different aspects.

Ability To relate the overall concept of subtraction to any of its embodiments.

In this topic we are concerned with recapitulating the 4 earlier aspects of subtrac-tion: taking away, comparison, complement, and change. Finally, in Activity 5, these are fused together into a concept of subtraction from which can be extracted all of these particular varieties.

activity 1 using set diagrams for taking away [Num 4.7/1] A teacher-led discussion for a small group. Its purpose is to relate the take-away

aspect of subtraction to set diagrams.

Materials • Pencil and paper for all.

1. Write on the left of the paper: 8 – 5 2. Sayanddraw(pointingfirsttothe8,thenthe5):

“This says we start with 8.”

“This says we take away 5.” Cross out the 5 to be taken away.

3. Write the result, 3. 4. Review the correspondences between the number sentence and the starting set,

the action (crossing out), and the result. 8 – 5 3 5. Let the children repeat steps 1 to 4 with another example. Use vertical notation,

as above. 6. Give further practice if needed.

Suggested sequence forthe discussion

Discussion of concepts

129

activity 2 using set diagrams for comparison [Num 4.7/2] A teacher-led discussion for a small group. Its purpose is to relate the comparison

aspect of subtraction to set diagrams, and thereby to what they have just done.

Materials • Pencil and paper for all.

1. Write as before. 8 – 5

2. Say, “This subtraction can have another meaning, besides taking away.” Here we have two numbers, the larger one above.

3. Draw these on the right of the subtraction sentence.

4. Ask (pointing): “How many more are there in this set, than this?” 5. If they answer correctly, say “Let’s check.” Draw lines like this and say (pointing) “These 5 lines show where the sets are alike, so these 3 without lines show where they are different.” 6. Iftheydonotanswercorrectly,usestep5toshowhowtheycanfindtheresult. 7. Either way, write the result in the subtraction sentence. 8. Review the correspondences between the number sentence, the two sets, the ac-

tion (comparison), and the result.

8 – 5 3

9. Explain: “We haven’t been taking away, so we shouldn’t read the number sen-tence as ‘take away.’ We say “8, subtract 5, result 3.”

10. Give further practice, as required. Use vertical notation only. 11. Say, “Now we have 2 meanings for this subtraction sentence.” Review these.

Suggested sequence forthe discussion

Num 4.7 Subtraction with all its meanings (cont.)

130

activity 3 Using set diagrams for finding complements [Num 4.7/3] A teacher-led discussion for a small group. Its purpose is to relate the complement

aspect of subtraction to set diagrams, and thereby to what they did in Activities 1 and 2. You may wish to review Num 3.5, ‘Complementary numbers,’ Activities 1 and 2, in the Addition network in SAIL Volume 1.

Materials • Pencil and paper for all. • Red and blue felt tips.

1. Write, and draw in pencil.

2. Say, “Here is another meaning. We’re told that these are all to be coloured red or blue. If 5 are coloured red, how many blue?”

3. If (as we hope) they say “Three,” check by colouring. If not, demonstrate. 4. Say, “If we didn’t have red and blue felt tips, what could we do instead?” 5. Accept any sensible answers, and contribute the suggestion below. Tell them

that they may continue to use their own way if they like.

6. Invite other meanings, e.g., 8 children, 5 girls, how many boys? 7. Review the correspondences between the number sentence, the whole set, and

the two parts of the set.

8 – 5 3

8. Remind them that the larger number has to be above. This time the larger number is the whole set, the next number is one part, and the last number is the other part.

9. Give further practice, as required. Use vertical notation only. 10. They now have 3 meanings for the subtraction sentence. Review these.

activity 4 using set diagrams for giving change [Num 4.7/4] A teacher-led discussion for a small group. Its purpose is to relate the ‘cash, cost,

change’ aspect of subtraction to set diagrams, and thereby to what they did in Activi-ties 1, 2, and 3.

Materials • Pencil and paper for all.

8 – 5

Num 4.7 Subtraction with all its meanings (cont.)

Suggested sequence forthe discussion

131

1. Write, on the left of the paper: 2. Say, “There’s just one more meaning we can give this. Suppose you are a shop-

keeper, and a customer gives you 8¢ for an apple. But the apple only costs 5¢. What money will you give back to him?”

3. Assuming that they answer correctly, say, “Yes. Now let’s check.” 4. Draw.

5. Say, “These are the 5 pennies for the apple,” and draw the partition line. Write the 5 inside.

6. Continue: “And so these are the pennies you give back to the customer.“ Point to the right-hand subset and write the 3 inside.

7. Relate the foregoing to ‘Cash, cost, change.’ (Num 4.6/3).

8. Give further practice, as required. 9. Review all 4 of the meanings they now have for subtraction.

activity 5 unpacking the parcel (subtraction) [Num 4.7/5] A game for up to 6 children. Its purpose is to consolidate children’s understanding

of the 4 different aspects of subtraction.

Materials • ‘Parcel’ cards, of two kinds, as illustrated in Stage (a) and Stage (b).* • A bowl of counters. * Provided in the photomasters

8 – 5

Num 4.7 Subtraction with all its meanings (cont.) Suggested sequence forthe discussion

132

Rules of Stage (a) the game 1. Thefirstsetofparcelcardsisputface down, and the top one turned over. (Reminder: this is read as “7, subtract 3, result 4,” NOT as “7, take away 3 . . . ”) 2. Explain that this has a number of different meanings which can be ‘taken out,’

one at a time, like unpacking a parcel. 3. The children take turns to give one meaning. If the others agree, the child whose

turn it is takes a counter. 4. There are four different mathematical meanings, as in Activities 1, 2, 3, and 4. 5. An unlimited number of situational meanings can also be found, and these can

become repetitive. E.g., if someone says, “7 boxes, 3 empty, so 4 have something in them,” and someone else then says, “7 cups, 3 empty, so 4 have something in them,” this is so little different as to be hardly worth saying. If the rest of the group

unanimously think that an example is of this kind, they might reject it even though correct. This might lead to discussion as to what is acceptable as a genu-inely different meaning.

Stage (b) 1. This is played in the same way as Stage (a), except that the second set of parcel

cards is used, like this one:

These have even more possible meanings, and the children should write these down before expanding them further. E.g.:

2. We thus have ‘parcels within parcels.’ This activity involves much concentra-tion of mathematical meaning, and children should return to it at intervals until all4aspectsaremastered.Childrenfindthepart-wholerelationshipsharderthan the take-away and comparison aspects of subtraction.

In this topic we have a good example of the highly abstract and concentrated nature of mathematical ideas. Hence its power, but hence also the need for very careful teaching. In the topics which lead up to this one, the 4 different aspects of subtraction are introduced separately, with the use of materials to provide a less abstract approach. In the present topic these are brought together by using set diagrams, which again provide a less abstract symbolism than the purely numerical symbols whose use, withfullunderstanding,isthefinallearninggoal. In Activity 5, the children are learning explicitly something about the nature of mathematics, namely its concentration of information. We ourselves have been taking notice of this from the beginning.

7– 3 4

Num 4.7 Subtraction with all its meanings (cont.)

4+ 3 7

7– 3 4

7– 4 3

Discussion of activities

133

Num 4.8 SUBTRacTiON OF NUmBeRS Up TO 20, iNcLUDiNG cROSSiNG The TeN BOUNDaRy

Concept Expansion of the subtraction concept to include larger numbers.

Ability To subtract numbers up to 20, including examples which involve crossing the 10 boundary.

‘Crossing the ten boundary’ means calculations like 12 – 3, 14 – 6, 17 – 9. All subsequent examples which involve regrouping, such as 52 – 4, 82 – 36,318 – 189, depend on this. Over the years there has been much discussion whether children should be taught to subtract by decomposition or complementary addition. The argument for decomposition has been that it can be demonstrated with physical materials, and so is better for teaching with understanding. For complementary addition, it has been claimed that it is easier to do, and makes for faster and more accurate calculations. Thelattercanalsobejustifiedsensibly,thoughitseldomis:‘borrowing’and‘pay-ing back’ is a nonsensical explanation. The present approach is based on the physical regroupings which the children have learned in Org 1, and the concept of canonical form which comes in both Org 1 and Num 2. It has several advantages: (i) It is mathematically sound. (ii)Itallowschildrentousewhichevertechniquetheyfindeasier:countingback, corresponding to the take-away aspect of subtraction, or counting on, cor-responding to complementation. Both of these they have already experienced with physical materials. Taking away across the ten boundary involves decomposing one larger group into 10 smaller groups; complementing involves putting together 10 smaller groups into one larger group. Either way, regrouping and canonical form are the key concepts. (iii) The same technique (changing out of or into canonical form) is good for adding, multiplying, and dividing. The present topic is preparatory to the full technique, which follows in Num 4.9. It teaches what we do after changing out of canonical form.

activity 1 Subtracting from teens: choose your method [Num 4.8/1] A teacher-led activity for up to 6 children. Its purpose is to show them two ways of

subtracting across the tens boundary, help them to see that these are equivalent, and choose which method they prefer.

Materials • Two sets of number cards, in different colours. One set is from 10-19, the other is from 0 to 9.*

• Subtraction board, as illustrated.* * Provided in the photomasters

Discussion of concepts

134

What they do 1. Beforestarting,theyshouldreviewfingercounting,including‘Teninmyhead.’(See Num 1.4/1, Num 1.5/1 and Num 1.7/1 in SAIL Volume 1.)

2. The subtraction board is put where all the children can see it the same way up. Bothsetsofcardsareshuffledandputfacedown,neartheboard,withtheteensset on the left.

3. The top card from each pack is turned over, and put one in each space on the board to give (e.g.)

4. Demonstratethetwowaysofdoingthisusingfingercounting. (a)Countingback.“Westartat13andcountbackto7,puttingdownonefinger

eachtime.12(onefingerdown),11,10,9,8,7(sixfingersdown).Thediffer-ence is 6.” They all do this.

(b)Countingon.“Westartat7andcountonto13,puttingdownonefingereachtime.8(onefingerdown),9,10,11,12,13(sixfingersdown).Thedifferenceis6.” They all do this.

5. Note that (i) We use the word ‘difference’ in both cases to link with this aspect of sub-

traction. (ii) We do notputdownafingerforthestartingnumber. (iii) This method gives (as intended) a mixture of examples in which some do

and some do not involve crossing the tens boundary. 6. With another pair of numbers, half the children arrive at the difference by count-

ing back, and half by counting on. They should all have down the same number offingers.

7. Step5isrepeateduntilthechildrenareproficient. 8. Tell them that having tried both, they may use whichever method they prefer

from now on. They may like to discuss the reasons for their preference.

Num 4.8 Subtraction of numbers up to 20 (cont.)

135

activity 2 Subtracting from teens: “check!” [Num 4.8/2] Agamefor4or6children,playinginteamsof2.Itspurposeistogivethemfluency

in subtracting across the tens boundary.

Materials • Number cards 10-19 and 0-9.* • Subtraction boards.* • A bowl of counters. *The same as for Activity 1.

What they do 1. A subtraction board is put where all can see it the same way up. Both sets of cardsareshuffledandputneartheboard,withthe‘teens’setontheleft.

2. Inthefirstteam,eachplayerturnsoverthetopcardfromoneofthepacksandputs it on the board, placing teens cards in the upper space.

3. The two players then do the subtraction independently by any method they like. 4. Another player says, “Ready? Check!” 5. Ontheword“Check,”bothplayersimmediatelyputfingersonthetabletoshow

their results. No alteration is allowed. 6. In some cases the result will be over 10. Example: 15 – 2. Both players should

nowputdown3fingers,saying“10inmyhead.”(SeeNum1.7/1inVolume 1.) 7. Theotherplayerscheck,andifbothhavethesamenumberoffingersontheta-

ble (and the others agree that this is the correct answer), this team takes a coun-ter.

8. Steps 2 to 6 are repeated by the next team. 9. Thegamecontinuesaslongasdesired,thenumbercardsbeingshuffledand

replaced when necessary. All teams should have the same number of tries. 10. The winners are those with the most counters.

activity 3 till receipts up to 20¢ [Num 4.8/3] A continuation from Num 4.6/3, for 3 to 6 children. Its purpose is to consolidate their

new skill in a familiar activity.

Materials • Play money. The customers each have four dimes as well as one nickel, and three pennies. The shopkeeper has a good assortment of all coins.

• A tray with partitions, used as a till. • Pictures on cards representing objects for sale with prices marked, ranging from

(say) 3¢ to 19¢. • Base 10 material, units and ten-rods.

What they do 1-7. The same as in the earlier version of ‘Till receipts’ (Num 4.6/3), except that the prices range all the way up to 19¢.

8. Customers may now purchase several objects at a time, provided only that the total is below 20¢.

9. Atanytimewhenchildrenhavedifficultyinwritingthetillreceiptsorcheckingthem, they should help themselves by using base 10 material. They may also use the counting on method of Num 4.6/2.

Num 4.8 Subtraction of numbers up to 20 (cont.)

136

activity 4 Gift shop [Num 4.8/4] A game, continuing on from the previous activity, for 3 to 6 children.

Materials The same as for Activity 3, together with a ‘notice’* as illustrated in step 9. Its pur-pose is further to consolidate their new skills, and extend these to subtraction other than from multiples of 10.

* Provided in the photomasters

Rules of play 1 - 8. The same as in Activity 3. 9. However, the shopkeeper also displays a notice:

10. If a customer thinks he has received incorrect change, the other customers also check. If it is agreed that the change was incorrect, the shopkeeper must give the customer his purchase free. The cash is returned to the customer, and the change to the shopkeeper.

11. If this happens 5 times the shopkeeper goes broke, and someone else takes over the shop. (The number of mistakes allowed to the shopkeeper should be adjust-ed to the children’s ability.)

12. Tomakethingsmoredifficultfortheshopkeeper,customersmaypaywithwhat-ever amounts they like. E.g., they could hand over 17¢ for an object costing 9¢. (This rule should not be introduced until the children have learned the rest of the game.)

It will be noticed that the activities of this topic do not begin with the use of physi-cal materials, in spite of the importance which in general we attach to these. Base ten material is good for teaching the exchange of 1 ten for 10 ones, but they al-ready have plenty of experience of this. It is also good for teaching conversion into and out of canonical form, and this too is done in earlier contributors to the present topic. It lends itself well to teaching subtraction in its ‘take-away’ form, but not nearly so easily to the comparison and complementation forms. The latter are easier to do mentally, since counting forward is easier than counting back. So the approach in this topic relies on the foundations laid by Mode 1 schema-build-inginearliertopics,andusesfingercountingasatransitionaltechniquewhichapplies equally well to either aspect of subtraction. This will fall into disuse as children gradually learn, and use for subtraction, their addition facts. These are followed by the application of these new techniques in familiar ac-tivities. The last activity, ‘Gift shop,’ introduces a penalty for the shopkeeper if he makes too many mistakes, and a reward for the customer who detects a mistake. It isnotonlyinthisgamethatashopkeeperwhocannotdohisarithmeticfindshim-selfindifficulties!

OBSeRve aND LiSTeN ReFLecT DiScUSS

Your PurcHaSE FrEE

if i give the wrong change.

Discussion of activities

Num 4.8 Subtraction of numbers up to 20 (cont.)

137

Num 4.9 SUBTRacTiON Up TO 99

Concept Their existing concept of subtraction, expanded to include subtraction of two-digit numbers.

Abilities (i) To subtract two-digit numbers. (ii) To apply this to a variety of situations.

So far as the concept itself is concerned, all that is new is the size of the numbers to which the operation is applied. However this requires the introduction of new techniques, and it is important that the manipulations of symbols which children learn at this stage should be meaningful in terms of the underlying mathematics. The method we recommend has been reached by much thought, discussion, andfieldtrialswithchildren.ItbeginsinNum4.8,andcontinueshere.However,rather than split the discussion, the whole of it was given at the beginning of Num 4.8. It would therefore be useful to reread this.

Before embarking on this, we need to guard against the common error of subtractingthe wrong way round, e.g.,

This gives the wrong result because subtraction is non-commutative. This con-trasts with addition, which is commutative. The result of these two additions is the same if we interchange the numbers:

Not so for these two subtractions:

cannot be done, with the numbers they know about so far

I leave it to your own judgment relative to the children you teach, whether or not to introduce the term ‘non-commutative’ at this stage. The important practical result is what Activity 1 is about.

Discussion of concept

32 – 17 25

7 2 – 2 – 7 5

7 2 + 2 + 7 9 9

138

Suggested sequence forthe discussion

activity 1 “can we subtract?” [Num 4.9/1] A teacher-led discussion for a small group, or for the class as a whole. Its purpose is

toemphasizethatonecanonlysubtractwhenthefirstnumberofthepairisgreaterthan or equal to the second. In vertical notation, the upper number must be greater than or equal to the lower. (At this stage we are not concerned with negative num-bers, but see the note at the end of this activity.)

Materials • Pencil and paper, or • Chalk and chalkboard.

1. Write a subtraction in vertical notation, e.g.,

2. Ask for the result. 3. Remindthemofthefirstofthemeaningsforsubtraction,takingaway. (See Num 4.7, ‘Subtraction with all its meanings’)

4. Reverse the numbers, and ask “Can we subtract this way round?”

5. Depending on their responses, let them see that this would mean making a set with number 2 and crossing out 6.

6. If there is still any doubt, let them try to do it with physical objects. 7. Repeat steps 1 to 6 with other examples. Include equal numbers, and also cases

where one number is zero. Start sometimes with the not-possible case. 8. Continue, “How about one of the other meanings of subtraction? Let’s try it

with cash, cost, change.” 9. Clearly in this case you get change.

10. But in this case, the shopkeeper would say, in effect, “Can’t be done.”

6 – 2

2 – 6

6 – 2 4

Num 4.9 Subtraction up to 99 (cont.)

cents cash 9 cost – 5 change 4

cents cash 5 cost – 9 change

139

11. Repeat steps 8 to 10 with one or more further examples. 12. Ask “What have we learned?” and obtain agreement on a suitable formulation,

in their own words. It should mean the same as the learning goal described in the heading for this activity.

Note Some children may say that they can subtract a larger number from a smaller (e.g., referring to digging a hole, or temperatures below zero, . . .). In this case you could tell them that they are quite right, but they are talking about a different kind of number system, called integers, which they haven’t come to yet. When they do, they willfindthatitstillusesamethodforsubtractionliketheonetheyarenowlearning.

activity 2 Subtracting two-digit numbers [Num 4.9/2] A teacher-led activity for a small group. Its purpose is to teach subtraction of two-

digit numbers, including cases which involve crossing the tens boundary. This now extends its meaning to include that between teens and twenties, twenties and thirties, etc.

Materials • Subtraction board.* • Two sets of number cards, in different colours. One set contains twenty cards

bearing assorted numbers between 50 and 99, the other contains twenty cards with assorted numbers all less than 50 (provided in the photomasters).**

* Like that for Num 4.8/1 ** Suitable assortments are: Under 50 – 8, 9, 12, 14, 17, 18, 21, 23, 26, 29, 30,

33, 36, 39, 41, 42, 44, 45, 47, 48. 50 and over – 50, 53, 56, 59, 61, 67, 68, 72, 74, 75,

78, 79, 80, 82, 84, 87, 91, 93, 96, 99.

1. Put out the subtraction board and number cards as in Num 4.8/1. Explain that this is like the earlier activity, but they are going to learn how to do subtraction of larger numbers.

2. Begin with an example such as this, in which both the upper digits are larger than the digits below them. 3. Explain: “We subtract a column at a time, ones from ones, tens from tens.” 4. Let them practice a few examples of this kind. 5. Next, introduce an example for which this is not so.

6. “Can we do ‘4 subtract 6’?” (No.) “So this is what we do. First, we write it in headed columns.”

Num 4.9 Subtraction up to 99 (cont.)

– 23

– 23

74 – 26

68

68

tens ones 7 4 – 2 6

Suggested sequence forthe discussion

140

Num 4.9 Subtraction up to 99 (cont.) 7. “Next, we write the upper number differently.” (This is changing it into a non-canonical form.)

8. “Now we can subtract a column at a time, as before.”

9. This is the answer in place-value notation.

10. The whole process may be set out as below. (The sign ⇔ means ‘is equivalent to.’ Its use is optional.)

This way they can see all the steps. At present the goal is understanding before speed.

11. The two middle steps may soon be combined mentally.

12. Now let the children use the subtraction board as in Num 4.8/1, to give assorted examples.Thatis,thetwopilesofnumbercardsareshuffledandputfacedown,and the top card from each pile is turned over and put on the board. The chil-dren copy what is there onto their papers, calculate the results, and check.

Notes (i) If step 7 is not understood, use base 10 material to demonstrate the exchange of 1 ten-rod for 10 cubes. The rest of the calculation is more easily dealt with symbolically,withthehelpoffingercountingifnecessary.Thisallowseitherthe use of counting forward from the smaller number (the complement aspect of subtraction), or counting back from the larger number (the take-away aspect of subtraction): see Num 4.8/1, ‘Subtracting from teens: choose your method.’ Manychildrenfindcomplementationtheeasiermethod.

(ii) Eventually, the intermediate steps may be done mentally, but this should not be done until children have had a lot of practice in the written form.

tens ones 6 14 – 2 6 4 8

74 – 26 48

74 – 26

74 – 26

⇔

tens ones 6 14 – 2 6 4 8

tens ones 7 4 – 2 6⇔ ⇔

74 – 26 48

tens ones 6 4 14 – 2 6 4 8

⇔7

74 – 26 48

⇔

tens ones 6 7 4 14 – 2 6

141

activity 3 Front window, rear window [Num 4.9/3] A game for two. Its purpose is to practise subtraction of two-digit numbers.

Materials • ‘Front window, rear window’ game board.* • ‘Car,’ as illustrated.* • Pencil and paper for each player. * Provided in the photomasters

The ‘car,’ in coloured cardboard, has windows (shaded areas) which are cut out as holes. The arrow shows the direction of travel.

What they do 1. This game is based on the fact that the road signs we see looking backward tell us distances to places we have left behind.

2. The players sit opposite each other with the board between them. 3. The car is put at the starting town (bottom left), with the arrows pointing in the

directionofmovement.Theplayersfindoutwhichwindowtheylookthroughfrom the writing they see right way up.

4. Thecarmovestothefirstroadsignandeachwritesthenumberhesees(say,87[km] through the front window and 27 [km] through the rear window).

5. The car moves on to the next road sign and each again writes the number he sees (say, 68 [km] through the front window and 46 [km] through the rear window).

6. Eachhasnowrecordedtwodistances.Bysubtraction,eachfindsthedistancethey have travelled between the road signs. Though the numbers are differ-ent, the distances are of course the same, so each passenger should get the same result.Forthefirsttwosignsinthisexample,thesubtractionsare:

7. The car moves on to the next road sign and steps 5 and 6 are repeated with the two latest numbers.

8. This continues to the end of the journey. At the intermediate towns the signs now relate to the next town ahead and the town just left. The car now begins to travel in the opposite direction, so the board needs to be turned around so that the passengers continue to face the same way relative to the car.

9. By changing seats on their next trip, the passengers could get a different set of calculations.

10. Furtherpracticemaybegivenbyusingotherfigures,eitherbymakingotherboards or by putting stickers on the existing board.

– 68 19

– 27 19

Num 4.9 Subtraction up to 99 (cont.)

4687

Front

Window

Window

Rear

and

142

Num 4.9 Subtraction up to 99 (cont.)

Front window, rear window [Num 4.9/3]

Anothersuitablesetoffiguresforthe‘Frontwindow,rearwindow’board:

Road Road section length 1 113 Front window 100 79 52 33 Rear window 13 34 61 80 2 125 Front window 99 76 51 36 Rear window 26 49 74 89 3 110 Front window 81 55 25 16 Rear window 29 55 85 94 4 100 Front window 78 57 38 19 Rear window 22 43 62 81

143

activity 4 Front window, rear window – make your own [Num 4.9/4] An activity and game for two. It is an extension of Activity 3, ‘Front window, rear

window,’ in which the children work out their own road signs. This is appreciably harder than Activity 3.

Materials • A board similar to that for Activity 3, without the numbers written in, but with the lines showing locations of the road signs. This board must be covered in transparentfilmorlaminated.

• ‘Car,’ as for Activity 2. • 2 dice, one marked 0-2 for tens, the other 1-6 for ones. • A washable non-permanent marker, and a damp rag.

What they do 1. Theyfirstdecideonthedistancebetweenthefirsttwotowns.Thismustbebe-tween 105 and 130 kilometres. They both write this number down.

2. They throw both dice, to obtain the distance travelled from the start. 3. One player marks in this distance for the ‘rear window,’ the other subtracts from

the total distance to get the ‘front window’ number and marks this on the board. 4. They throw the dice again for the next distance travelled. 5. They each work out their own number, and mark it on the board. 6. Steps4and5arethenrepeatedtwicemoretocompletethefirststageofthejour-

ney. 7. They then repeat steps 1 to 6 for the remaining stages of the journey, choosing a

different distance between each pair of towns. 8. They should then play as in Activity 3 to check their calculations, or swap

boardswithanotherpairtodothis.Atfirsttheymightprefertocheckattheendof each stage of the journey.

Activity 1 is intended to prevent the error of subtracting the wrong way around, discussedfirstinthe‘Discussionofconcepts’forthistopic. Activity 2 then shows children how to subtract two-digit numbers. Since chil-dren should already be familiar with moving in and out of canonical form from the addition network, the suggestion is that they work at this level rather than go back to physical embodiments in base ten material. This avoids going right back to the take-away aspect of subtraction, whereas the concept now includes other compo-nents. Help from the latter may however be used to demonstrate the change from canonical form, also (correctly) called decomposition and (incorrectly) called bor-rowing, without detriment to the foregoing. Activities 3 and 4 are more sophisticated applications of the concept of subtrac-tion than they have encountered so far, using the new technique which they have learned.

OBSeRve aND LiSTeN ReFLecT DiScUSS

Num 4.9 Subtraction up to 99 (cont.)

Discussion of activities

144

Num 4.10 SUBTRacTiON Up TO 999

Concept The existing concept of subtraction, extrapolated to numbers up to 999.

Abilities (i) To subtract numbers up to 999. (ii) To apply this to situations involving any of the four aspects of subtraction al-

ready encountered.

Although all that is new here is (as in the previous topic) an expansion of the size of numbers to which the operation of subtraction is applied, this results in a sub-stantial increase in the amount of information to be handled. This makes necessary careful organizing, with the help of symbols on paper.

activity 1 race from 500 to 0 [Num 4.10/1]

An activity for 2 children. Its purpose is to make sure that the relation between larger numbers and physical materials, and the process of exchanging, are kept active be-fore the children embark on the symbolic work of Activity 2.

Materials • Base 10 materials in box: 5 ten-squares at least 20 ten-rods at least 20 unit cubes • 2 dice 1-6 • Two hundreds, tens, ones boards, as illustrated below, one for each player.

Rules of 1. Each player starts with 5 ten-squares, representing 500. These are put in the the game hundreds space on his board. 2. They throw the two dice alternately, and these determine the number to be

subtracted. If the numbers thrown are (e.g.) 2 and 6, the player may choose to subtract either 2 tens and 6 units, or 6 tens and 2 units.

Discussion of concept

Hundreds OnesTens

145

8 2 5- 2 4 1 5 8 4

7 12 5 - 2 4 1 5 8 4

H t ones 8 2 5 - 2 4 1 5 8 4

H t ones

3. They physically take away that number in the base ten material, exchanging with the material in the box where necessary.

4. It is important to return material not in use to the box, or confusion results. 5. Theplayerwhofirstreacheszeroisthewinner. 6. Zero must be reached exactly. To do this, players may add or subtract the num-

bers thrown. Both must be used. E.g., a player throwing 5 and 2 could subtract either 7 or 3, but not 5 or 2.

7. Either player may delay taking his turn in order to check what his opponent is doing.

activity 2 Subtracting three-digit numbers [Num 4.10/2]

A teacher-led activity for a small group. It follows on from Num 4.8/1, and its pur-pose is to extend the children’s ability to subtract to three digit numbers.

Materials • Subtraction board.* • Two sets of number cards, in different colours. One set contains twenty cards

bearing assorted numbers between 500 and 999, the other contains twenty cards bearing numbers all less than 500.**

* Like that for Num 4.8/1 ** Suitable assortments provided in the photomasters: Underfivehundred:36,54,60,71,115,142,179,187,200,241,254,298,323,332,

387, 399, 406, 415, 423, 468. Overfivehundred:5l4,533,576,587,606,641,652,688,751,727,765,790,825,

843, 879, 890, 900, 914, 939, 968.

Suggested 1. Put out the subtraction board and number cards as in Num 4.8/1 and 4.9/2. sequence for Explain that they are now going to continue their learning of subtraction to the discussion hundreds, tens and ones. 2. As before, the top card from each pile is turned over and put on the subtraction

board. The children copy onto their paper. 3. The method is a direct continuation of that already learned, so you may invite

their own suggestions as to how to deal with these new examples. 4. Changing out of non-canonical form may now involve one column, or two. Example: one column involved.

This 584 is written last.

Num 4.10 Subtraction up to 999 (cont.)

146

9 1 4- 2 9 8 6 1 6

9 0 14 - 2 9 8

H t ones 9 1 4 - 2 9 8

H t ones 8 10 14 - 2 9 8 6 1 6

H t ones

9 0 0- 1 4 2 7 5 8

8 10 0 - 1 4 2

H t onesH t ones 8 9 10 - 1 4 2 7 5 8

9 0 0 - 1 4 2

Example: two columns involved.

This 616 is written last.

Not all examples will be as hard as the second; but even with this kind, and the one below, this technique keeps thinking under control. It reduces the amount which has to be dealt with to one step at a time, and still allows children to do these mentally when they feel ready.

Example: two zeros.

This 758 is written last.

5. If one of the harder kinds comes up before the children are able to deal with it, it maybeby-passedsothatchildrenbuildupconfidencewitheasierexamples.

6. Ifchildrenstillhavedifficulty,theyprobablyneedmorepracticechangingfromcanonical form with the use of physical materials. (See Num 2.11.)

7. Steps 2 to 4 are now repeated. Each child copies from the board onto his paper, and does the subtraction. They then compare results with those sitting next to them.Theyshouldcontinuepracticingontheirownuntilallareproficient.

activity 3 airliner [Num 4.10/3]

A game for two children. Its purpose is to illustrate the practical use of subtraction, and give practice in the subtraction of three-digit numbers.

Materials • Airliner board, see Figure 7.* • Three packs of 0-9 number cards.* • Pencil and paper for each player. * Provided in the photomasters

What they do 1. Oneplayeractsasflightengineer,theotherasfueloperator. 2. Theysitoppositeeachotherwiththeboardinbetween,facingtheflightengi-

neer.Somekindofscreenisarrangedsothatonlytheflightengineercanseetheaircraft’s fuel gauges (as would be the case in the actual situation).

3. Thenumberpacksareputtogetherandshuffled. 4. The 6 top cards are put in the 6 spaces. This gives a simulation of the readings

on the fuel gauges when the aircraft lands.

H t ones

Num 4.10 Subtraction up to 999 (cont.)

147

5. Theflightengineerwantstotakeonboardtheexactamountoffueltoequalizeher port and starboard tanks. Suppose that these are the readings.

PORT TANK STARBOARD TANK 724 419

She calculates that she wants 305 gallons in her starboard tank, and asks the fuel operator to put this in.

Figure 7 Airliner

6. Thefueloperatorasks,“Whatisyourpresentreading,please?”Theflightengi-neer tells her.

7. The fuel operator calculates what the reading should be when he has pumped in 305 gallons (in the present case). This of course is 724, the reading on the port tank gauge which he cannot see.

8. The fuel operator says, “Check, please. I have put in 305 gallons according to my own gauge, so yours should now read 724.”

9. Ifbothcalculationsarecorrect,thisfigurewillbethesameasthereadingontheport tank gauge (which the fuel operator cannot see). If not they look for the mistakes in their calculations.

10. Steps4to9arerepeated,simulatinganotherflight. 11. Afterafewflights,theplayerschangeroles.

Num 4.10 Subtraction up to 999 (cont.)

Starboard Fuel tank Port Fuel tank

148

activity 4 candy store: selling and stocktaking [Num 4.10/4]

An activity for 4 to 6 children. Its purpose is to consolidate their new abilities in a game which links it back again with physical embodiments.

Materials • ‘Candies.’ Whatever candy-sized objects you can obtain. • Small self-sealing plastic bags. • Cardboard boxes to hold ten bags. • A pack of 30 spending cards on which are written sums of money varying be-

tween 4¢ and 25¢.* • Play money. • Pencil and notebook for each child. • Shallow money containers, one for each player (useful but not essential). * Provided in the photomasters

What they do 1. One acts as shop owner, one as shop assistant, and the rest as customers. 2. Before opening the shop, the shop owner and shop assistant pack the candies

into bags of 10. They also pack boxes containing 10 bags. 3. The players begin with about 100¢ each, and the shop has about 200¢ (not these

exact amounts, however). 4. Before the shop opens, the shop owner checks and records his stock and cash:

e.g., stock 347 candies, cash 185¢. 5. Likewise each customer notes how much money he starts with. 6. Shoppingnowbegins.Eachcandycosts1¢.Thecustomersfindithardtode-

cide how much to spend on candies, so each in turn takes the top card from the pile of spending cards which is put face down on the table.

7. Having taken a card, the customer spends that amount on candies, keeping the card as a record of what he has spent.

8. The shop assistant does the selling. She breaks bulk as may be necessary to give customers the number of candies they ask for. She also gives change if required.

9. This continues until each customer has made (say) 3 purchases. With 4 custom-ers, this will make 12 transactions.

10. At closing time, the shop owner checks stock and cash. The total value of the candies still in stock, added to the money taken, should be equal to the stock at the beginning of the day.

11. Meanwhile the customers also check. The total of their spending cards would be equal to the value of their candies. Also, the money they have spent subtracted from what they started with should be equal to the amount they have left.

12. Another pair then takes over as shop owner and shop assistant. Everyone helps in restoring all the materials to their starting positions, and another day’s shop-ping can begin.

Note Whenchildrenareproficientatthis,thepricepercandymaybevaried.Thiscomplicates matters considerably. It might be a good activity for use with calcu-lators.

Num 4.10 Subtraction up to 999 (cont.)

149

Activity 1 returns to the use of Mode 1 schema building, to make sure that the symbolic manipulations do not lose their connections with their underlying con-cepts. Base ten material embodies grouping and exchanging particularly well, so we use it once again here. Activity 2 extends the pencil-and-paper technique for subtraction, already learned for numbers up to 99, into the domain of three-digit numbers. This is a simple form of Mode 3 building - extrapolating ideas and skills they already have into new situations. It is simple, because no new concepts are involved. Activity 3 is an easy game in which the players have a good reason for checking each other’s calculations. Activity 4 is a fairly ambitious one which brings together all the aspects of sub-traction described earlier in this network. There is taking away, since the candies sold are taken from those in the shop. At any time, the value of the candies in stock and the money taken are complementary, the total being the starting value of the stock before any is sold and before any money is taken. This is also true for the candies in stock and the money in the till. Giving change is also involved, and the change due is the difference between the cost of the goods and the cash tendered. The predictions are made and tested, in the checking of stock and tak-ings, money spent and candies acquired. And it is likely that at some stage of the activity children will decide to use recording to help them in what they are doing. There is a lot in this package, so it will be worth while letting children repeat it for as many times as they still enjoy it, so that they may have time to consolidate the many relationships which are embodied.

OBSeRve aND LiSTeN ReFLecT DiScUSS

Discussion of activities

Num 4.10 Subtraction up to 999 (cont.)

Candy store: selling and stocktaking [Num 4.10/4]

150

Number stories (multiplication) [Num 5.4/1]

151

Discussion of concepts

[Num 5] Multiplication combining two operations

Num 5.4 NUMBER STORIES: ABSTRACTING NUMBER SENTENCES

Concept Numbers and numerical operations as models for actual happenings, or for verbal de-scriptions of these.

Abilities (i) To produce numerical models in physical materials corresponding to given number stories, to manipulate these appropriately, and to interpret the result in the context of the number story: first verbally, then recording in the form of a number sentence.

(ii) To use number sentences predictively to solve verbally given problems.

The concept of abstracting number sentences is that already discussed in SAIL Volume 1 in Num 3.4, and it will be worth reading this again. In Num 4.4 it was expanded to include subtraction, and here we expand it further to include multipli-cation. In some applications of multiplication, we need to place less emphasis on the first action and the second action, and more on their results: namely a small set (of single objects) and a large set (a set of these sets). The use of small set ovals and a large set loop from the beginning provides continuity here.

activity 1 number stories (multiplication) [Num 5.4/1] An activity for 2 to 6 children. Its purpose is to connect simple verbal problems with

physical events, linked with the idea that we can use objects to represent other ob-jects.

Materials • Number stories on cards, of the kind in the example overleaf. Some of these should be personalized, as in Num 3.4/1 in Volume 1, but now there should also be some which do not relate to the children, more like the kind they will meet in textbooks. Also, about half of these should have the number corresponding to the small set coming first, and about half the other way about.**

• Name cards for use with the personalized number stories. • Number cards 2 to 6.** • 30 small objects to manipulate: e.g., bottle tops, shells, counters, . . . • 6 small set ovals. * • A large set loop.* • Slips of blank paper, pencils. * As for Num 5.1/1 in SAIL Volume 1. ** Provided in the photomasters

152

What they do (apportioned according to how many children there are) 1. A number story is chosen. The name, cards and number cards are shuffled and

put face down. 2. If it is a personalized number story; the top name card is turned over and put in

the number story. Otherwise, explain “Some of these stories are about you, and some are about imaginary people.”

3. The top number card is turned over and put in the first blank space on the story card.

4. The next number card is turned over and put in the second blank space. 5. The number story now looks like this.

6. Using their small objects (e.g., shells) to represent children, the oval cards for boats, and the set loop for the lake, the children together make a physical repre-sentation of the number story. If it is a personalized number story, this should be done by the named child.

7. The total number of shells is counted, and the result written on a slip of paper. This is put in the space on the card to complete the story.

8. While this is being done, one of the children then says aloud what they are do-ing. E.g., “We haven’t any boats, so we’ll pretend these cards are boats, and put shells on them for children. We need 3 children in each boat, and 4 boats inside this loop which we’re using for the lake. Counting the shells, we have 12 chil-dren boating on the lake.”

9. The materials are restored to their starting positions, and steps 1 to 8 are repeated.

activity 2 abstracting number sentences [Num 5.4/2] An extension to Activity 1 which may be included fairly soon. Its purpose is to teach

children to abstract a number sentence from a verbal description.

Materials • As for Activity 1, and also: • Pencil and paper for each child.

What they do As for Activity 1, up to step 8. 9. Each child then writes a number sentence, as in step 6 of Num 5.3/1. They read

their number sentences aloud. 10. The materials are restored to their starting positions, and steps 1 to 8 are repeated.

Num 5.4 Number stories: abstracting number sentences (cont.)

There are 3 children in each rowing boat

and 4 rowing boats on the lake. So

altogether children are boating on the lake.

153

Num 5.4 Number stories: abstracting number sentences (cont.)

Giles is collecting fir cones in a wood.He gets 5 cones into each pocket and hehas 6 pockets.When he gets home and empties hispockets, how many fir cones will he have?

activity 3 number stories, and predicting from number sentences [Num 5.4/3] An activity for 2 to 6 children. It combines Activities 1 and 2, but in this case com-

pleting the number sentence is used to make a prediction as in Num 5.3/2.

Materials • Number stories asking for predictions.** • Name cards for use with the personalized number stories.* • Two sets of number cards 2 to 6.** • 30 objects to manipulate. * • 6 oval small-set cards. * • A set loop. * • Pencil and paper for each child. * *As for Activity 1. ** Provided in the photomasters What they do (apportioned according to how many children there are) 1. A number story is chosen. The name cards and number cards are shuffled and

put face down. 2. If it is a personalized number story, the top name card is turned over and put in

the number story. Otherwise, explain “Some of these stories are about you, and some are about imaginary people.”

3. Two more children turn over the top two number cards, and put these in the first two spaces on the story card.

4. The number story now looks something like this:

5. The named child (if there is one) then puts out an appropriate number of small-set cards, and a number card from the second pack, to represent the situation. (This corresponds to step 6 of Activity 1.) For the present example, this is what he should put.

154

6. He explains, using his own words and pointing: “These 6 ovals represent my 6 pockets, and this 5 is the number of fir cones I have in each.” (This verbaliza-tion is an important part of the activity.)

7. All the children then write and complete number sentences, as in Activity 2. For the present example, these would be

either 6(5) = 30 “six fives make 30” or 5 x 6 = 30 “five, six times, equals 30” They read these aloud. 8. Another child then tests their predictions by putting the appropriate number

of objects in each small-set oval. The number sentences are corrected if necessary.

9. They then write their answers to the question as complete sentences. E.g. (in this case), “Giles will have 30 fir cones.” 10. The materials are returned to their starting positions, and steps 1 to 9 are repeat-

ed using a different number story and numbers.

The activities in this topic parallel those in Num 3.4 (addition) and Num 4.4 (sub-traction), and in view of their importance it will be worth rereading the discussion of these. Solving verbally stated problems is one of the things which children find most difficult, as usually taught. This is because they try to go directly from the words to the mathematical symbols. In the early stages, it is of great importance that the connection is made via the use of physical materials, since these correspond well both to the imaginary events in the verbally-stated problem, and to the mathemati-cal schemas required to solve the problem. The route shown below may look longer, but the connections are much easier to make at the present stage of learning.

Since these physical materials have already been used in the earlier stages for schema building, they lead naturally to the appropriate mathematical operations. Side by side with this, they learn the mathematical symbolism which will in due course take the place of the physical materials.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Num 5.4 Number stories: abstracting number sentences (cont.)

155

Num 5.5 MULTIPLICATION IS COMMUTATIVE; ALTERNATIVE NOTATIONS; BINARY MULTIPLICATION

Concepts (i) The commutative property of multiplication. (ii) Alternative notations for multiplication. (iii) Multiplication as a binary operation.

Abilities (i) To understand why the result is still the same if the two numbers in a multiplica-tion sentence are interchanged.

(ii) To recognise and write multiplication statements in different notations with the same meanings.

(iii) To multiply a pair of numbers.

(i) Particularly when considered in a physical embodiment, the commutative property of multiplication is interesting, and surprising, if we come to it with fresh eyes. Why should 5 cars with 3 passengers in each convey the same number of persons as 3 cars with 5 passengers in each? And likewise whatever the numbers? If you think that the answers are obvious, try to explain the second of these before reading further. I have tried to introduce this element of surprise in the first two activities.

(ii) So far the children may have used only one notation for multiplication but there are several in common use.

5(3) which may be read as “Five times three” or “Five threes.” 3 x5 read as “Three, five times.” 3 x 5 read as “Three multiplied by five” (note the equal spacing) which may be taken either to mean the same as the one above, or as representing binary multiplication.

There is also this 3 vertical notation x 5

which will be needed later for calculations 473 like this x 5

We are more likely to read the lower one as “five threes . . . five sevens [seventies]. . . five fours [four-hundreds] . . .” than as “three, five times . . . seven [seventy], five times . . . four [four hundred], five times.” So here is an inconsistency. This inconsistency is neatly removed by using the notation for binary multiplication, which is explained in the next section.

(iii) Consider three different multiplications: 5(3) 6(3) 7(3)

Discussion of concepts

156

The operand in every case is 3, but there are three different operations. These are, unary multiplication by 5, by 6, and by 7. In contrast, binary multiplication has a pair of numbers as operand; and there is just one operation, multiply, for all pairs of numbers. All the foregoing can be taken care of at a single stroke, by introducing compu-ter notation for multiplication.

3 ∗ 5 = 15 read as “Three star five equals fifteen”

means the binary product of 3 and 5. Because multiplication is commutative (see Activity 2), it includes and replaces all the following:

5(3) = 15 3 x5 = 15

3(5) = 15 5 x3 = 15

3 5 x 5 x 3 15 15 This seems good value to me, particularly since children need to learn computer notation anyway. The computer books do not specifically mention binary multipli-cation: I am including this meaning as a bonus. All the above will become clearer as you work through the activities in this topic.

activity 1 Big Giant and little Giant [Num 5.5/1]

An activity for up to six children. It is a sequel to ‘Giant strides on a number track’ (Num 5.1/3 in SAIL Volume 1), and its purpose is to introduce the commutative property of multiplication in a way which does not make it seem obvious (which it is not).

Materials • Activity board, see Figure 8.* • Two sets of number cards 6 to 9 (single).* • One set of number cards 2 to 5 (double).* • Double width number track 1 to 50.* • Blu-tak or Plasticine. * Provided in the photomasters The number cards need to fit the spaces in the activity board. See illustration follow-

ing. The squares on the number track need to be accurate in size, 1 cm by 1 cm, since the

track will also be used with 1 cm cubes in Activity 2.

Number cards

Num 5.5 Multiplication is commutative; alternative notations; binary multiplication (cont.)

2

26

157

What they do 1. Two children act the parts of Big Giant and Little Giant, respectively. 2. Big Giant has the set of double number cards and one of the sets of single

number cards. Little Giant has the other set of single cards. 3. Big Giant turns over the top cards in each of his packs, and puts these face up in

their respective spaces on the activity board. 4. Big Giant puts into action the statement which he has completed. He does this

by putting footprints (small blobs of Plasticine) on the number track, one for each stride of the given length. He uses the upper part of the number track.

5. Little Giant’s stride is of a different length. He has to find out how many of these strides he must take to arrive at the same place as Big Giant. He does this in the same way as Big Giant, using the lower part of the number track.

6. He then puts the appropriate number card in his second space to complete his own statement.

7. The children then read aloud from the board what they have done. 8. Steps 3 to 6 are now repeated, with other children taking the parts of Big Giant

and Little Giant. Say to Little Giant: “If you think you know how many strides you need, to get to the same place as Big Giant, put the number in the space first and then see if you were right. If you don’t know, find out first and then put in the number.”

9. Eventually one of the children acting as Little Giant will realize that his own numbers are always the same as Big Giant’s the other way around. He will then be able to predict successfully every time. He keeps this discovery to himself for the time being.

10. This continues, with children taking turns at Little Giant, until all have discov-ered how to predict, though probably not yet why.

11. When this stage is reached, ask: “But can you explain why your method always works?”

12. If any Little Giant can give a good explanation before having done Activity 2, then he is really clever!

Num 5.5 Multiplication is commutative; alternative notations; binary multiplication (cont.)

Figure 8 Big Giant and Little Giant

158

activity 2 little Giant explains why [Num 5.5/2] A teacher-led discussion for up to 6 children. Its purpose is to provide the explana-

tion asked for in step 11 of Activity 1. For this, it may be necessary for you to take the part of Little Giant.

Materials • The same as for Activity 1, and also • Squared paper and pencil for each child. • 100 1 cm cubes, if possible 20 each in 5 different colours.

What they do 1. Start with a set of cards in position on the activity board, as in steps 3, 4, 5 of Activity 1. This time, Little Giant’s card should also be face up. E.g.:

Big Giant: “My stride is 7 spaces and I shall take 3 strides.” Little Giant: “My stride is 3 spaces and I shall get there in 7 strides.” 2. Instead of footmarks on the number track, the giants represent their strides by

rods. So in this example, Big Giant uses a 7-rod to represent one of his strides, and puts together 3 of these on the number track. Little Giant uses a 3-rod for one of his strides, and joins together 7 of these alongside those of Big Giant. Adjacent strides should be of different colours, to keep them distinguishable.

3. At this stage it is still not obvious why both are of the same length, since the two journeys look different.

4. “Now,” says Little Giant, “we arrange the rods like this.” The long rods are then separated again into single strides, and arranged in rectangles as shown below.

5. From this it can be seen why the two journeys are of the same length. 3 strides, each of 7 spaces makes the same rectangle as 7 strides, each of 3 spaces. Both rectangles have the same number of cubes.

6. Little Giant asks, “Will it always happen like this, whatever the numbers?” 7. The group as a whole discusses this. One or two other examples might be done

using the cubes. 8. This should now be continued as a pencil and paper activity, using squared pa-

per, as follows. 9. Two more numbers are shown on the activity board: e.g., 6 and 4. 10. All the children draw rectangles 6 squares long and 4 squares wide. 11. The children make these into diagrams for the two different ways of getting to

the same place, as shown in step 4. About half the children make diagrams rep-resenting Big Giant’s journey (6 spaces in a stride, 4 strides), and the rest make diagrams representing Little Giant’s journey (4 spaces in a stride, 6 strides).

12. They interchange, and check each other’s diagrams. 13. Finally, Little Giant tells the others: “There is a word for what we have just

learned. Multiplication is commutative. It means that if we change the two numbers around, we still get the same result. Always.”

Num 5.5 Multiplication is commutative; alternative notations; binary multiplication (cont.)

3 strides

3 spaces in a stride

little Giant Big Giant

7 spaces in a stride 7 strides

159

activity 3 Binary multiplication [Num 5.5/3] Continuing the teacher-led discussion in Activity 2.

Materials • Pencil and squared paper.

1. Draw a rectangle, say 5 squares by 3 squares

2. Ask the children how many different multiplication sentences they can write

based on this diagram. Collect these in two groups, those meaning 5 sets of 3 and those meaning 3 sets of 5. (See ‘Discussion of concepts.’)

3. Now that they have learned that multiplication is commutative, all the notations in both groups are different ways of writing and thinking about the same thing. So it would be sensible to have just one way of writing all these.

4. Computer notation provides just what we need.

3 ∗ 5

combines all the above in just one notation. So we shall use this from now on, unless there is a particular reason for using one of the others.

5. (This part of the discussion is optional ) It follows that 5 ∗ 3 and 3 ∗ 5 mean the same thing as each other. It would be nice to have a notation in which neither number was written before the other, but it is hard to think how this could be done. The nearest representation we can get to this is the rectangle 3 squares by 5 squares, shown in step 1. Here, there is no distinction between rows and columns; i.e., no distinction between the 3 and the 5.

activity 4 unpacking the parcel (binary multiplication) alternative notations [Num 5.5/4]

A game for up to 6 children. Its purpose is to consolidate the connections between the notation just learned, and its various possible meanings. Also, to show what a great amount of information is contained in a single statement.

Materials • About 10 cards on which are written an assortment of number sentences like 3 * 5 = 15, 6 * 4 = 24.†

• Pencil, plain and squared paper for each child. • Number track (e.g., the double width number track from Num 5.5/1). • A bowl of counters.

.

Num 5.5 Multiplication is commutative; alternative notations; binary multiplication (cont.)

Suggested sequence for the discussion

† Provided in the photomasters

160

Discussion of activities

Num 5.5 Multiplication is commutative; alternative notations; binary multiplication (cont.)

What they do 1. The cards are shuffled and put face down. 2. The top card is turned over: e.g., 2 ∗ 7 = 14. 3. The players in turn ‘unpack the parcel,’ by giving some of the various mean-

ings contained in this number sentence. They may use drawings, or appropriate number sentences.

E.g., (drawing) “Make a set of 2. Make it 7 times. Combined result, 14”

writes: 2 x 7 = 14

4. The next player might draw this, and say “Rectangle 2 by 7, 14 squares. The notation for this is the one on the card.”

5. The next players might use the same rectangle, first as 7 2-rods, then as 2 7-rods, writing these respectively as 7(2) = 14 and 2(7) = 14.

6. Points may also be scored by writing a number sentence in a different notation, which has the same meaning as that which another player has just given. This may be done only once for each example.

7. Others might use the number track, in the ways already learned. 8. Each time a player gives a meaning, the others say “Agree” if they think he is

correct, and he takes a counter. If they do not agree, they explain why not and if necessary discuss.

9. When all have made a statement about this card, or when players run out of ideas, the next card is turned, and steps 2 to 6 are repeated.

10. The player with the most counters at the end of play is the winner.

In this topic, the children are introduced to some more ways of symbolizing multiplication. For their future work they will need to know all of these. To minimize the likelihood of confusion and maximize the advantages de-scribed, the important thing is to continue to strengthen the connections between the symbols and the concepts, rather than between symbols and each other.

Not this: symbols symbols

symbols symbols

but this: symbols symbols symbols symbols

concepts (schema) The best way to do this is by frequent use of physical materials and drawings.

OBSERVE AND LISTEN REFLECT DISCUSS

161

Num 5.6 BUILDING PRODUCT TABLES: READY-FOR-USE RESULTS

Concepts (i) Product tables, as an organized collection of ready-for-use results. (ii) The complete set of products, up to 10 ∗ 10.

Abilities (i) To recall easily and accurately whatever results are needed for a particular job. (ii) To build new results from those which are already known.

In view of the strong emphasis throughout this book on concepts, schemas, and understanding, it may come as a surprise that I believe firmly in the importance of children ‘knowing their tables,’ even in these days of inexpensive calculators. But there is no inconsistency. To put our knowledge to good use, which includes using it to extend its own boundaries, we must have readily available all the results which we need for fre-quent use. Just as our writing would be very slow if we had not memorized the correct spelling of a large number of words, so our arithmetic would be very slow if we had not learned our addition facts and product tables. However, we would hardly think it sensible to make children memorize the spelling of words which to them were meaningless. Yet until recently, and perhaps even now, children are required to memorize multiplication tables while their con-cept of multiplication is so weak that they still have to ask, “Please, Teacher, is it an add or a multiply?” Understanding of the relationship between multiplication and physical events, both with actual objects and events as described in number stories, has been care-fully developed in preceding topics. In the present topic, although the activities are intended to help children acquire a repertoire of ready-for-use results, you will see that this is done in such a way that many of these are built up from results already known. So children are learning, not a collection of isolated facts but a system of interrelated results. Activity 1 also uses an important property of multiplication, called by mathematicians the distributive property: “multiplication is distributive over addition.” E.g.,

The use of the product patterns shows this clearly, and at this stage of children’s learning this intuitive and pictorial understanding is enough. A statement like the above would only confuse. Later, we shall use this same property to multiply by numbers greater than 10. E.g., to multiply by 58, we multiply by 50 and by 8 and add the results. Still later, this property will be much used in algebra. a ∗ (y + z) = a ∗ y + a ∗ z

Discussion of concepts

4 ∗ 8 = 4 ∗ (5 + 3) = 4 ∗ 5 + 4 ∗ 3

162

activity 1 Building sets of products [Num 5.6/1]

An activity for children in pairs. Its purpose is to help children build sets of product results; and to know these, ready for use. Stages (a) and (b) also give practice in mental addition.

Materials • 4 sets of product patterns, for the products of 2, 3, 4, 5 respectively. * • 4 corresponding sets of symbol cards. * * Provided in the photomasters. See examples in step 3 below.

What they do Stage (a) Products to 5 1. Each pair uses one set of product patterns, and the first half (∗ 1 to ∗ 5) of the

corresponding set of symbol cards. We will suppose that they are using the sets of products of 4, so the symbol cards will be those from 4 ∗ 1 to 4 ∗ 5 inclusive.

2. The product patterns are laid face up in order, and the symbol cards are shuffled and put face down.

3. Each child in turn turns over the top symbol card, puts it below the correspond-ing product pattern, and says what this product is. (That is, the total number of dots in the rectangle.) E.g., she picks up 4 ∗ 3, puts it as shown below,

and says “4 star 3 is 12.” 4. She may obtain this result either by counting the dots, or by recalling it from

memory. 5. The other player then says either “Agree” or “Disagree.” If “Disagree,” they

check by counting the dots. 6. The symbol card is then replaced in the pack, face down as before. 7. It is important that steps 3 to 6 are repeated until the children can recall all the

results from memory before they continue to Stage (b). Stage (b) Products to 10 1. A set of product patterns is put in a row, each with its symbol card just below.

(See illustration in step 3, above.) 2. The other half of the symbol cards are now used. As before they are shuffled

and put face down, and the top one is turned over. Suppose that it is 4 ∗ 8. 3. To deal with this, the product patterns for 4 ∗ 3 and 4 ∗ 5 are put together, each

with its symbol card, as shown on the next page.

Num 5.6 Building product tables: ready-for-use results (cont.)

Set of products of 4

4 ∗ 3symbol card

163

4. The player then says:

“4 ∗ 8 is 4 ∗ 3 plus 4 ∗ 5 12 plus 20 4 ∗ 8 is 32”

5. The symbol card is put on one side, and the product rectangles put back in line. 6. This is continued until all the symbol cards have been used. This activity is

repeated until the children are fluent.

Notes (i) Stage (b) is best introduced with even sets of products, 2 ∗ and 4 ∗. These give multiples of 10 for the products 2 ∗ 5 and 4 ∗ 5.

(ii) The products with 10 should come late in the sequence. Let children devise their own responses. Some double the product with 5, some put together 3 product patterns, some already know the result without a product pattern.

activity 2 “i know another way.” [Num 5.6/2]

An activity for children in pairs. Its purpose is to consolidate the results already known, to make further connections between these, and to extend their use of the distributive property of multiplication over addition. (This activity often arises spon-taneously during Activity 1, Stage (b).)

Materials • A double set of product patterns.* • Symbol cards, full sets.* *As used in Activity 1, Stage (b).

4 ∗ 8

4 ∗ 2 4 ∗ 44 ∗ 1

4 ∗ 3 4 ∗ 5

Num 5.6 Building product tables: ready-for-use results (cont.)

164

1, 2, 3. As in steps 1, 2, 3 of Activity 1, Stage (b). In this case there are two similar rows of product rectangles. Note that in Step 2, the beginning player may now use either a single product rectangle if available (e.g., for 4 ∗ 3) or two of these (e.g., 4 ∗ 2 + 4 ∗ 1).

4. If the other player agrees, she says “Agree.” 5. She then says “I know another way.” Using again the example given in Activity

1, Stage (b), Step 3, and following on from there, she might put together product rectangles like this

and says “4 ∗ 8 is 4 ∗ 4 plus 4 ∗ 4 16 plus 16 4 ∗ 8 is 32 as before.”

6. If the first player agrees, she says “Agree.” 7. The cards are replaced, and steps 1 to 6 are repeated. 8. As the players progress, they may use 3 product rectangles if they wish.

activity 3 completing the product table [Num 5.6/3]

An investigative activity for up to 6 children. The amount of help needed from their teacher will depend on the ability of the children.

Materials For each child: • A partly-completed product square, from 1 to 50.* • An L-card, see illustration on next page.* • Pencil. * A full-size product square is provided in the photomasters, SAIL Volume 2a.

Rules of the game

Num 5.6 Building product tables: ready-for-use results (cont.)

4 ∗ 4 4 ∗ 4

4 ∗ 8

165

Stage (a) What is the connection between the numbers in the squares, and the product patterns

and results they have learned so far? (The answer is shown, in brief, in the illustra-tion.)

Stage (b) How can they complete the product table by filling in the blank squares? (The left

hand column, of course, continues from 5 to 10). We then want all the products of 5, of 6, of 7, of 8, of 9, and of 10. One way would be by laying down the L shape for each product, and counting squares. This is not as bad as it sounds, since it only involves counting the squares additional to those already numbered. But there are better ways.

(i) The first column, of course, continues to 10. (ii) The next three columns can be completed by using their knowledge that multi-

plication is commutative. So 6 ∗ 2 = 2 ∗ 6 which is already known, and so on. (iii) Row 5 can be completed by counting by 5’s. [5, 10, 15, 20, . . .] (iv) Row 6 can now be completed by adding 6 each time. This is in fact another use

of the distributive property.

And so on.

(iv) The children should write all their new results lightly in pencil, until all their figures are checked with each others’. Then they should complete their prod-uct table as neatly as possible in ink (e.g., ballpoint, thin fibre-tip) for their own future use. Tell them to keep these carefully.

What they are asked to findout

The lower right-hand number gives the number of squares in the rectan-gle, which is the same as the number of dots in the corresponding product pattern.

Here we have:

4 ∗ 7 = 28

6 ∗ 6 = 6 ∗ 5 + 6 ∗ 1 = 30 + 6 = 36

Num 5.6 Building product tables: ready-for-use results (cont.)

1 2 3 4 5 6 7

2 4 6 8 10 12 14

3 6 9 12 15 18 21

4 8 12 16 20 24 28

166

activity 4 cards on the table [Num 5.6/4]

An activity, for children to play in pairs, as many as you have materials for. They may with advantage make their own, and practice in odd times which would other-wise be wasted. Its purpose is to practice the recall of all their product results.

Materials • 9 sets of symbol cards, each with 10 cards in each set, from 2 ∗ 1 to 10 ∗ 10.† • One product table and L-card for each pair.† † Provided in the photomasters

What they do 1. In each pair, one child has in her hand a single pack of cards, shuffled and face down. The other has on the table her multiplication table and L-card.

2. Child A looks at the top card, say 3 ∗ 8, and tries to recall this result. Child B then checks by using her multiplication square and L-card.

3. If A’s answer was correct, this card is put on the table. If incorrect, it is put at the bottom of the pile in her hand so that it will appear again later.

4. A continues until all the cards are on the table. This method gives extra practice with the cards she got wrong.

5. Steps 1 to 4 are repeated until A makes no mistake, and her hand is empty. 6. The children then change roles, and repeat steps 1 to 5. 7. Steps 1 to 5 are then, possibly at some other time, repeated with a different pack

until all the packs are known. 8. The final stage is to mix all the packs together. Each child then takes from these

a pack of 10 mixed cards, and repeats steps 1 to 5 with this pack. 9. Step 7 is then repeated with a different pack. 10. This activity should be continued over quite a long period: say, one new pack a

week, with review of earlier packs, including mixed packs.

activity 5 products practice [Num 5.6/5] A game for up to 6 children. Its purpose is further to consolidate children’s recall of

multiplication results. This game may be introduced for variety before children have completed Activity 4, using the packs which they have learned so far.

Materials • Multiplication cards: all the packs which they have learned, mixed together.* • A multiplication table and L-card.* • Products board, see Figure 9.* * Provided in the photomasters

Rules of play 1. All, or nearly all, the cards are dealt to the players. Each should have the same number, so when the remaining cards are not enough for a complete round, they are put aside and not used for the game. The products board is put on the table between them.

2. The players hold their cards face down. In turn they look at their top card (e.g., 7∗8) and put it in the appropriate space on the products board (in this case 56). It does not matter if there is a card in that space already – the new card is then put on top.

3. The others check. If it is wrong, they tell her the correct answer and she replaces the card at the bottom of the pack.

Num 5.6 Building product tables: ready-for-use results (cont.)

167

4. If she does not know, she asks and someone tells her. She then replaces the card at the bottom of the pack.

5. Any disagreements are settled by using the multiplication board. 6. Play continues until all have put down all their cards. If there are no mistakes,

all will finish in the same round. Those who do make mistakes, or do not know, will be left with cards in their hands to put down in subsequent rounds.

7. If one player finishes a clear round ahead of the others, she is the winner.

Variation If a stopwatch is available, this game may also be played as a race. To make a fair race, each player needs to be using the same pack. This suggests various forms, e.g.,

Form (a) A single pack, of a table to be consolidated or reviewed. Form (b) Several packs mixed. Form (c) (For advanced players.) All packs from twos to tens, making 90 cards in all.

The rules for all forms are the same: 1. One player acts as starter and timekeeper. 2. The others in turn see how quickly and accurately they can put down all their

cards. 3. Those not otherwise involved check for accuracy, after all the cards have been

put down. 4. For each incorrect result, 5 seconds are added to the time. (This figure may be

varied according to the skill of the players.) 5. The winner is the player with the fastest time after correction for errors.

Num 5.6 Building product tables: ready-for-use results (cont.)

= 7 = 8 = 9 = 10 = 12 = 14

= 15 = 16 = 18 = 20 = 21 = 24

= 25 = 27 = 28 = 30 = 32 = 35

= 36 = 40 = 42 = 45 = 48 = 49

= 50 = 54 = 56 = 60 = 63 = 64

= 70 = 72 = 80 = 81 = 90 = 100

Figure 9 Products board

= 2 = 3 = 4 = 5 = 6 products practice

168

activity 6 Multiples rummy [Num 5.6/6]

A popular game of the rummy family. Best for 2 to 5 players; 6 is possible. Its pur-pose is to consolidate their knowledge of multiplication facts.

Materials • Two packs of double-headed number cards, a beginners’ pack and an advanced pack.* The beginners’ pack containing two each of all the numbers from 2 to 30, excluding primes greater than 20 (54 cards). The advanced pack contains one each of all the numbers from 2 to 100, excluding primes greater than 40 and even numbers greater than 20 (43 cards).

* Provided in the photomasters

1. The packs are put together and shuffled. Five cards are dealt to each player. 2. The rest of the pack is put face down on the table, with the top card turned over

to start a face upwards pile. 3. The object is to get rid of one’s cards by putting down sets of 3 multiples of the

same number; e.g., 3, 15, 21; 2, 18, 20. 4. Players begin by looking at their cards and putting down any trios they can.

They check each others’ trios. 5. The first player then picks up a card from either the face down or the face up

pile, whichever she prefers. If she now has a trio, she puts it down. Finally she discards one of her cards onto the face-up pile.

6. In turn the other players pick up, put down if they can, discard. (This sequence should be memorized.)

7. The winner is the first to put down all her cards. Play then ceases. 8. The others score the total of the remaining cards in their hands. The lower the

score the better, the winner scoring zero. 9. Another round may then be played, and the scores added to those of the previous

round. The overall winner is the one with the lowest total score.

Num 5.6 Building product tables: ready-for-use results (cont.)

Rules of the game

Multiples rummy [Num 5.6/6]

169

{

The emphasis of this topic is to combine easy and accurate recall of multiplication results (products) with knowledge of many of their interrelationships. This means that although there is memorizing to be done, the facts which are thus learned are related both to each other, and to their underlying numerical concepts. It also means that if a particular result is temporarily forgotten, the children know ways of reconstructing it for themselves. Thus in Activity 1, the products are related to patterns of dots which show both

the numbers to be multiplied and their product

Numerals are not written on the product pattern cards, so that children do not just match symbols to symbols in this activity. The number of rows and columns are however both small enough to be subitized – perceived without counting. In Stage (b), known products are used to construct new ones. This is Mode 3 concept building, creativity. Here they are also building up the interconnections mentioned above. Practice in mental addition is also provided – though you may allow them to use pencil-and-paper while gaining confidence. Activity 2 further builds up the interconnections, and consolidates use of the distributive property. Activity 3 is another constructive, extrapolative activity: Mode 3 again. Activities 4 and 5 are for developing effortless recall – ‘facts at their fingertips.’ But with such a strong relational beginning in Activities 1, 2, and 3, this should not become just rote learning. What we want children to acquire is fluency: easy recall without loss of meaning. In Activity 4, it is tempting to write the results on the back of the cards; e.g., to write 18 on the back of 6 ∗ 3. I suggest that it is better not to do this, since it is a step in the direction of rote memory.

It strengthensthis link symbol symbol

whereas we want this symbol symbol

schema

Eventually we symbol symbolwant all three links schema

and Activity 5 promotes learning of the link shown in dashes. But the other two links, shown in continuous lines, are the most important. Activity 6, multiples rummy, is a popular game which uses recall of relevant products to make the best decisions.

OBSERVE AND LISTEN REFLECT DISCUSS

5

{3

Num 5.6 Building product tables: ready-for-use results (cont.)

Discussion of activities

170

Num 5.7 MULTIPLYING 2- OR 3-DIGIT NUMBERS BY SINGLE DIGIT NUMBERS

Concept Expansion of the multiplication concept to include examples in which the larger number has 2 or 3 digits, the smaller still having only 1 digit.

Ability To calculate any result of the kind described.

It is no small thing that once we know the multiplication facts from 1 ∗ 1 to 9 ∗ 9, we can use these to multiply numbers of any size, such as 5283 ∗ 364. Now that calculators are so widely available, it is no longer necessary for children to achieve a high degree of proficiency in doing calculations of this kind, and there are some teachers who think that these should now be omitted altogether. My personal view is that this ability to make use of a relatively small number of multiplication facts (55, including products with 1 and 10) to multiply any two numbers we choose, without learning any more multiplication tables, is a fine example of extending our knowledge from within. Also, it depends on certain properties of multiplication and addition which, together with the idea of a variable, form the main foundations of algebra. So I continue here along the path towards long multiplication, in the belief that there are interesting things to be learned on the way. There are five of these properties altogether, of which two are important for this ex-pansion of multiplication. They have been given technical names by mathematicians. (i) Multiplication is distributive over addition.

If we think of 40 as 4 tens, we can calculate this because we know 4 ∗ 6. We already know 7 ∗ 6. The distributive property says we can get the correct result by adding these two products.(ii) Multiplication is associative.

You can easily check that this is not true for division. Its importance is that it al-lows us to multiply by one factor at a time. E.g.:

which we can calculate using the distributive property. (Note that we have also used a shortcut for multiplying by 10, which needs to be justified.)It is not easy to decide how much of this should be explained to the children. A full explanation is probably too much for most, but to teach just the methods is to teach instrumentally something which calculators can do much better. So what I have done is to present the distributive property intuitively, as em-bodied in base 10 material; and the associative property explicitly, followed by a demonstration of its usefulness. In both cases I have replaced the technical names by simpler descriptions, but some of your children may like to know the former.

For example, 47 ∗ 6 = (40 + 7) ∗ 6 = (40 ∗ 6) + (7 ∗ 6)

E.g, 2 ∗ 3 ∗ 4 will give the same result whether the first two, or the second two, are multiplied first: (2 ∗ 3) ∗ 4 = 6 ∗ 4 2 ∗ (3 ∗ 4) = 2 ∗ 12 = 24 = 24

38 ∗ 40 = 38 ∗ (10 ∗ 4) = (38 ∗ 10) ∗ 4 = 380 ∗ 4

Discussion of concept

171

activity 1 using multiplication facts for larger numbers [Num 5.7/1]

An activity for any number of children working in pairs. Its purpose is to take the first steps in extending children’s ability to multiply. Before they do this activity, it may be desirable to review canonical form (Num 2.11).

Materials • Base 10 material, tens and units. • 6 small set ovals.* • Large set loop. • Two 1-to-6 dice for each pair. • Two 1-to-9 dice for each pair. • Pencil and paper for each child. * Two are provided in the photomasters

What they do 1. The 1-to-6 dice are used to begin with. First two are thrown together, to give a two digit number. Then a single die is thrown to give a single digit number.

2. These are then written by each pair as a multiplication, in vertical notation, using headed columns. For example:

3. Base 10 material is put in one of the small set ovals to represent 43.

4. They then put more small set ovals to make six in all, and a set loop around them. (Base 10 material is only used in one of the small set ovals.)

5. They think and write as follows. “We have to make 6 sets like the one shown. 6 sets of 3 units is 18 units. 6 sets of 4 tens is 24 tens.

Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers (cont.)

4 3 ∗ 6

H t ones

4 3 ∗ 6 24 18

H t ones

172

Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers (cont.)

H t ones

43 ∗ 6 = 258

4 3 ∗ 6 2 5 8

6. Rearrange in canonical form (working mentally from right to left), and there is our answer.

7. They compare their final answers. If these are different, they discuss. 8. Steps 1 to 6 are repeated. 9. The use of the base 10 material may be discontinued when you think that the

children have well established the concept that what they are doing on paper represents making many sets like the one in the set loop.

10. When the children are proficient with these smaller numbers, they may change to the two 1-to-9 dice.

11. The notation shown in steps 5 - 6 leads naturally to the conventional notation, in which step 5 is omitted and the rearrangement into canonical form is done as we go along, with help from small ‘carrying’ figures. This conventional notation is quick and convenient, but it is also very condensed. The notation in step 5 shows much more clearly what we are doing, so I recommend using this to start with. Note that the condensed notation will be needed by the time they reach long multiplication in Num 5.10.

activity 2 Multiplying 3-digit numbers [Num 5.7/2]

An activity for any number of children working in pairs. It follows directly on from Activity 1, but the larger number now goes up to 999.

Materials As for Activity 1, but with base 10 materials in hundreds, tens, and ones, and 3 dice for each pair of children.

What they do This is so like Activity 1 that it does not need to be described in detail. The method already learned is easily extended to these larger numbers.

activity 3 cargo boats [Num 5.7/3]

This game is best for 2 or 3 players, though more could play. Its purpose is to give plenty of practice in multiplication.

Materials • Game board, as in the photomasters and illustrated in Figure 10. • Die 1 - 6 (for easier game), 1 - 9 (for harder game). • Cargo boat for each player as in the photomasters. • Loading list for each player, as in the photomasters. • Calculator.

The cargo boats are cut out of cardboard and look like this.

400 300 500 200100

173

Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers (cont.)

1. Each player has a cargo boat, which he sails to each of 5 islands picking up cargo.

2. On each island there are packages of varying weight. 3. The first player sails his boat to the first island, and throws the die. This tells

him the number of packages which he must take. 4. Although the number of packages is given by the roll of the die, he may choose

the kind of packets he takes. All packets of the same kind must go into the same hold. The capacity of these is shown both on the boat and on the loading list (see next step).

5. Each player makes a loading list, on which he records what he takes on board. If he threw a five at the first island, his loading list might then look like this.

6. The first player then sails his boat to the second island and repeats the process. Meanwhile the next player sails to the first island and starts loading; and so on. This means that by the third turn, all three players are occupied in making their calculations.

7. The winner is the player who returns to the home port with the heaviest cargo. 8. Players use a calculator to check each other’s calculations.

Rules of the game

compart- packages load ment

number Weight

100

200

300 5 ∗ 50 250

400

500

total cargo

174

Figure 10 Cargo boats

Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers (cont.)

175

Activity 1 is a good example of Mode 3 schema building: the extrapolation of schema and technique to include larger numbers. In doing this, we want the ex-trapolation to be based on expanding the concept itself, and not just repeating a way of manipulating symbols in a different situation. The latter takes place too, but the justification for doing it is provided by the base-10 material. This takes us back to the concept of multiplication: start with a set, make it many times. Now our set has tens and units, but the way to make many of it is visually obvious. And it is this which leads to, and justifies, the written method. Here as elsewhere, headed column notation combined with the concept of ca-nonical form is an important way of representing what we are doing symbolically. Since the vertical lines are there to act as separators, there is no loss of meaning if we leave out the outside lines, left and right. And although I have still included the headings Th, H, T, Ones, I would see no objection to letting these gradually be-come, as it were, ‘unwritten headings.’ The extension to hundreds, tens, ones is so straightforward that I saw no reason for postponing it. Hence Activity 2. Activity 3 is another game like ‘Air freight’ and ‘One tonne van drivers,’ for practising calculations in an interesting situation. You will easily be able to de-vise others. Catalogue shopping also lends itself easily to purchasing several of a given article, and hence to price calculations which require multiplication. I think that we should at this stage allow children to use a calculator for check-ing their work. In the present situation I see calculators as labour-saving devices, rather than as completely removing the need for children to learn to do these calcu-lations themselves.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers (cont.)

Discussion of activities

Cargo boats [Num 5.7/3]

176

Suggested sequence for the discussion

Discussion of concept

th H t ones

Num 5.8 MULTIPLYING BY 10 AND 100

Concept The mathematics behind the well-known shorthand.

Abilities (i) To multiply by 10 or 100, using the shorthand discussed below. (ii) To explain it in terms of place-value.

If asked “How do you multiply by 10?”, many children will say “You add a zero.” This of course is not true. Adding zero to any number leaves it unchanged. Writ-ing a zero after the last digit is a useful shortcut, but try using it to multiply 2.5 by 10 and it will give a wrong answer. Short-cuts can lead you astray if you don’t know how they work.

activity 1 Multiplying by 10 or 100 [Num 5.8/1]

A teacher-led discussion for up to 6 children. Its purpose is to relate the ‘shorthand’ method of multiplying by 10 or 100 to its meaning.

Materials • Two 1-to-9 dice. • Base10 material. • Base 10 Th H T Ones board.* • Pencil and paper for each child. * As illustrated below in step 3. This needs to be fairly large.

1. Throw the two dice to give a random 2-digit number; say, 37. 2. Have them put out base 10 material on the board to represent this. In the present

example:

177

Num 5.8 Multiplying by 10 and 100 (cont.)

th H t ones

∗ 10

3. Ask them to show what this would look like if multiplied by 10, using additional base 10 material in the space below.

4. Ask them to record the result.

37 ∗ 10 = 370

5. Repeat steps 1, 2, 3 using different numbers. 6. Repeat step 1, and ask if they can predict the result. If anyone talks about “add-

ing a zero,” help them to see that this is incorrect by asking them what they get by adding (say) 3 + zero.

7. What we are really doing is moving every figure one place to the left, so that the units become the same number of tens, the tens become the same number of hundreds, and so on.

178

activity 2 Explaining the shorthand [Num 5.8/2]

This is a teacher-led discussion for up to 6 children. It replaces Activity 1 for chil-dren who already know the shorthand, but do not understand what is behind it.

Materials • As for Activity 1.

1. Use the dice to give a random 2 digit number as before, and let them use their shorthand to multiply by 10.

2. Ask them to explain why this works, using the same explanation (that they are not adding a zero) as in Activity 1 step 6.

3. If they give an explanation like that in Activity 1, step 7, or any other which is mathematically correct, good. Otherwise produce the base 10 Th H T Ones board, and take them through steps 2, 3, and 7 of Activity 1. Repeat, if neces-sary, until you are satisfied that they understand what they are doing.

activity 3 Multiplying by hundreds and thousands [Num 5.8/3]

A continuation of Activity 1 or Activity 2. First review Num 2.13/2 ‘Naming big numbers’ in the original form. Then repeat it

using zeros to the right of a 2 digit number. Then, using zeros to the right of a 3 digit number.

This is a topic in which the use of the short cut without understanding is wide-spread. Activities 1 and 2 therefore use a strong combination of headed columns and base 10 materials to provide a sound conceptual foundation for what is un-doubtedly a useful shortcut. Activity 3 parallels these at the symbolic level, em-phasizing the changes in meaning of each digit which result from their different positions.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 5.8 Multiplying by 10 and 100 (cont.)

Suggested sequence for the discussion

Discussion of activities

179

Num 5.9 MULTIPLYING BY 20 TO 90 AND BY 200 TO 900

Concept The combination of their new knowledge of how to multiply by 10 and by 100 with their existing multiplication tables.

Abilities To multiply any two or three digit number (later, more) by 20 to 90 and by 200 to 900.

The next step is easy. To multiply, say, 47 by 30 we multiply by 10 and by 3 in either order. What makes this true is the associative property, mentioned in Num 5.7. This tells us that

47 ∗ 30 = 47 ∗ (10 ∗ 3) = (47 ∗ 10) ∗ 3 = 470 ∗ 3

and for the last we only need to know our 3 times table.

activity 1 “How many cubes in this brick?” (alternative paths) [Num 5.9/1]

An activity for up to 6 children. Its purpose is to introduce children to the associa-tive property.

Materials • About 100 2 cm cubes. • Pencil and paper for each child.

What they do 1. Some of the children make a solid 2 by 3 by 4 brick. (Call it a cuboid if you prefer.) Others make bricks of different dimensions, according to how many children and how many cubes there are.

2. Using the first brick, ask how many small cubes (or units) are there, and how they arrived at this result.

3. There are 3 possible paths, depending on which layer they calculate first.

6 ∗ 4

2 ∗ 3 ∗ 4 2 ∗ 12 24

3 ∗ 8

4. Elicit all three of these, and record them as above. 5. Discuss what this shows, namely that when multiplying three numbers together

the result is the same whichever pair we combine (associate) first. 6. Help them to interpret these three results in relation to the brick. E.g. for the top

path, 2 ∗ 3 was calculated first. This corresponds to a layer, 2 cubes by 3 cubes, having 6 cubes in all. 6 ∗ 4 means that there are 4 such layers.

7. Ask whether they think that this is always true, whatever the numbers. (It is: the number of cubes in a brick of any size can be calculated in all of these ways.)

Discussion of concept

180

Num 5.9 Multiplying by 20 to 90 and by 200 to 900 (cont.)

activity 2 Multiplying by n-ty and any hundred [Num 5.9/2]

An activity for up to 6 children. Its purpose is to combine the shorthand just learned with the associative property, in order to multiply by 20, 30, . . . , n-ty (where n is any number from 2 to 9); and likewise for 200, 300, . . . , any hundred.

Materials • Two 1-to-9 dice. • Pencil and paper for each child.

What they do Stage (a) 1. Begin with numbers which do not give rise to special difficulties, such as extra

zeros. E.g., 87 ∗ 40. 2. Explain that by writing this as:

87 ∗ 40 = 87 ∗ 10 ∗ 4

we can get the result without knowing our 40 times table, or our 87 times table. We just need to know how to multiply by 10, and then by 4. (Activity 1 tells us that we will get the same result this way. Think of a brick 87 by 10 by 4.)

3. The work may be set out as below. 87 ∗ 40 = 87 ∗ 10 ∗ 4

Th H T Ones

8 7

∗ 10 = 8 7 0

∗ 4 = 32 28 0

= 3 4 8 0 [Canonical form]

87 ∗ 40 = 3480

4. Remind them that when converting into canonical form we work from right to left. When the new method is understood, the conventional notation may be reintroduced. (See Num 5.7/1, step 11.)

5. This is what we have now done.

We started here This path was too difficult 87 ∗ 40

87 ∗ 10 ∗ 4 3480

870 ∗ 4

So we went the other way around, which was easier.

181

Num 5.9 Multiplying by 20 to 90 and by 200 to 900 (cont.)

6. Even though this path is easier, there is a lot going on here, and it is important at this stage to consolidate these new ideas by discussion and by doing plenty of examples.

7. Repeat with other numbers. 8. When the children are confident at multiplying by numbers like 30 or 80, let

them extend the method to multiplying by numbers such as 300 or 800. Stage (b) In preparation for long multiplication, they multiply simultaneously by 10 and by 4,

as below. This may be taken in three steps.

1. 87 ∗ 40 = ____

We have now taken the first step into long multiplication, and this topic offers yet another example of how much is condensed into what on the surface looks like a simple process. I would not expect children necessarily to remember in detail all the reasons which they have been shown. What I think is important is for them to remember that there are reasons which they understood as they worked through them. They know that they did not learn rules without reasons, and that if neces-sary – possibly with a little help – they could remember why the new methods are correct. Although the children have already learned to change mentally into canonical form in topic 5.7/1, step 11, we back-track a little into the headed column notation because this notation is less condensed, and shows the meaning more explicitly. In other words, we are refreshing the connections between the methods and the underlying concepts.

OBSERVE AND LISTEN REFLECT DISCUSS

Th H T Ones

8 7

∗ 40 = 32 28 0

= 3 4 8 0

2. 87 ∗ 40 = 3480

Th H T Ones

8 7

∗ 40 = 3 4 8 0

3. 87 ∗ 40 3480

Discussion of activities

182

A suggested teaching approach

Num 5.10 LONG MULTIPLICATION

Concept The use of just 55 known results (from 2 ∗ 1 to 10 ∗ 10) to calculate any desired product.

Ability To multiply two numbers in which the smaller has 2 or 3 digits, with understanding and accuracy but not necessarily with speed.

In the past, the main emphasis has usually been on the technique of long multi-plication. The wide availability of inexpensive calculators has so changed the situation that if all that mattered was getting the answer, there would no longer be any good reason to teach this technique. This may well be the case for less able children. And I think that calculators should be freely used by all children in cases where the result is what is important, or when the figures are difficult. However, the validity of the technique of long multiplication depends on sev-eral important mathematical principles, which have already been mentioned and which continue to be important in later mathematics. Two of these have already been specifically dealt with, in topic 5 (multiplication is commutative) and topic 9 (multiplication is associative). A third has been well used, and made explicit for teachers but not for pupils, in topic 6 (multiplication is distributive over addition). There are two more which perhaps I should mention for completeness, but you do not need to think about them unless you want to. These are, that addition also is both commutative and associative. These properties make it possible to use known results to construct new knowl-edge, and thus to exercise the creative function of intelligence. This seems to me a good reason for continuing to teach long multiplication to those children who can grasp the principles on which it is based; and perhaps it is the only good reason. The contributory concepts and abilities have been carefully prepared in earlier topics, and it is now simply a matter of putting them together.

activity 1 long multiplication [Num 5.10/1]

For any number of children. Its purpose is to show them how methods and results which they already know can be combined to multiply any two numbers.

Materials • Pencil and paper for each child. • Any source of numbers to be multiplied, such as dice, number cards, text book.

1. Explain that having learned 55 multiplication results – called products for short –they can now multiply any two numbers they want, without having to memo-rize any new product tables. Discuss this in contrast to other learning situations. (Having learned the names of 55 people does not enable us to know the right name for every other person we will ever meet.)

Discussion of concept

183

2. Recall how they worked out new products from those they already knew. E.g., 4 ∗ 7 = 4 ∗ 5 + 4 ∗ 2 = 20 + 8 = 28

3. Show a suitable introductory example, such as 87 ∗ 43, and invite suggestions. 4. What do they already know how to work out? Accept and list all suggestions

which might be useful. 5. Depending on your own judgement of the ability of the children, you may wish

to leave them time for their own investigations, or you may decide that a direct exposition is appropriate at this stage.

6. There are a number of ways of setting out the work. Here are two which show clearly the principles involved.

(a) 87 ∗ 43 = 87 ∗ 40 + 87 ∗ 3 87 87 3480 ∗ 40 ∗ 3 + 261 3480 261 3741

87 ∗ 43 = 3741 (b) 87 ∗ 40 = 3480 87 ∗ 3 = 261 87 ∗ 43 = 3741

7. I would regard the transition to the conventional, condensed layout as optional. It saves little time, and does not show so clearly the mathematical principles used.

8. Give the children further practice in these until the method is well established. 9. You could explain that calculators are a quick and convenient way of doing cal-

culations like these; but it is good to be independent of calculators, and also to understand the mathematics behind the result.

activity 2 treasure chest [Num 5.10/2]

A game for two crews, each of up to 3 children. Its purpose is to give practice in us-ing long multiplication to make the best decisions.

Materials • ‘Treasure chest’ game board, see Figure 11.* • Sets of treasure cards, 4 to a set.∗† • Pencil and paper for each child. † One of these is illustrated. Six more sets of suitable numbers are given at the top of

the second following page. If you prefer, you may vary the objects according to your own imagination. These figures have been calculated to give close results, for which estimation is not sufficient. This makes it necessary to use long multiplication.

* Provided in the photomasters

Num 5.10 Long multiplication (cont.)

184

Figure 11 Treasure chest

Num 5.10 Long multiplication (cont.)

Her

e is

a c

lose

-up

of w

hat i

s w

ritte

n on

the

ches

ts.

32

gar

nets

45

am

ber b

rooc

hes

38 je

t bea

ds

36 a

met

hyst

s

w

orth

w

orth

w

orth

w

orth

$

74 e

ach

$ 53

eac

h $

64 e

ach

$ 67

eac

h

185

Suitable numbers for treasure cards:

Amber Jasper beads Garnets Jet First set 43 & 75, 59 & 56, 93 & 35, 39 & 64

Amethysts Topaz Gold Rubies Second set 82 & 67, 86 & 64, 47 & 117, 24 & 230

Silver goblets Emeralds Garnets Doubloons Third set 24 & 166, 33 & 139, 56 & 79, 31 & 145

Amber Amethysts Pearls Diamonds Fourth set 96 & 78, 56 & 134, 36 & 208, 26 & 278

Diamonds Jet Sapphires Garnets Fifth set 26 & 249, 225 & 29, 44 & 148, 94 & 69

Jasper Diamonds Pearls Amber Sixth set 175 & 54, 28 & 338, 37 & 256, 263 & 36

1. The two crews of treasure seekers have swum under water to the cave, where they find four small sealed chests.

2. The contents are painted on the outside. 3. To avoid conflict, they reach the following agreement. (i) Each crew will take back two chests. (ii) One crew will have first and fourth choice, the other will have second and

third choice. (Why is this the fairest way?) 4. They determine which crew will have which pair of choices by guessing in

which hand a pebble is held. 5. Since they have no calculator with them, they have to use long multiplication to

work out the values of the chests. 6. It should be left to them to realize that it is better for the crew which guesses

correctly in step 4 to delay saying whether they will choose first until after they have worked out the values of the chests.

7. In turn, as described in step 3, they take their chests and swim back to the boat. 8. On arriving on board, they may check with a calculator to find out which team

has done best. 9. The game may be repeated with other sets of chests.

Activity 1 involves using a combination of concepts and methods already learned to extend their ability to multiply into a much larger domain. When this method has been learned, Activity 2 provides a game in which to consolidate it. There is something further to be said about calculators. What these manipulate with such speed, convenience, and accuracy are not numbers but numerals – not mathematical concepts, but symbols for these. It is their users who attach mean-ings to the symbols. So the more we use calculators, the more important it is to keep sight of the meaning of what we are doing.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 5.10 Long multiplication (cont.)

Rules of the game

Discussion of activities

186

Different questions, same answer. Why? [Num 6.3/1]

187

[Num 6] Division sharing equally, grouping, factoring

Num 6.3 DIVISION AS A MATHEMATICAL OPERATION

Concept The connection between grouping and sharing.

Ability To explain this connection. (This is most easily done with the help of physical mate-rials.)

Physically, grouping and sharing look quite different.

Start with 15.

Make groups of 3.

Resulting numberof groups is 5.

Start with 15.

Share among 3.

Number in eachshare is 5.

It is only at the level of thought that we can see that these are, in a certain way, alike. When children have grasped the connection between grouping and sharing, they have the higher order concept of division.

Activity 1 Different questions, same answer. Why? [Num 6.3/1] A problem for children to work at in small groups. (I suggest twos or threes.) The

purpose is for them to discover for themselves the connection between grouping and sharing.

Materials • Two-question board, see Figure 12.* • Start cards 10 to 25.* • Action cards 2 to 5.* • 50 (or more) small objects. • Pencil and paper for each child. * Provided in the photomasters

Discussion of concept

188

Figure 12 Different questions, same answer. Why?

Num 6.3 Division as a mathematical operation (cont.)

189

Introducing 1. Thestartandactioncardsareshuffledandputfacedownintheupperspaceson the problem the two question board. 2. The top card of each pile is turned and put face up in the lower space. 3. Byreadingaboveandbelowtheline,therearenowtwounfinishednumbersen-

tences. E.g.,

4. One (or more) in each group copies down the upper sentence, and one (or more) the lower sentence. They then complete whichever sentence they have written, using physical objects if they like. E.g.,

N.B. They should write these neatly, and keep them for later use in Activity 2. 5. They draw diagrams to show what they have done. These diagrams could be

used in step 4, instead of physical objects. E.g.,

Grouping

Sharing

6. They compare the two results. 7. Steps 2, 3, 4 are repeated. Step 5 need not be repeated every time: its purpose is

to emphasize that these are two different questions. 8. Ask, “Will the two results always be the same? If so, why?” 9. Leave them to discuss this, and to arrive at a clear explanation. (One suggestion

will be found in the ‘Discussion of activities.’) 10. Return and hear their explanation, discussing it if necessary.

Num 6.3 Division as a mathematical operation (cont.)

190

Activity 2 Combining the number sentences [Num 6.3/2] An activity for a small group. Its purpose is to teach the notation for the mathemati-

cal operation of division.

Materials • The number sentences which they have written in Activity 1. • Pencil and paper for each child.

What they do 1. Havethemcomparethefirstnumbersentenceofeachkind.E.g.,

2. Tell them that these two meanings may be combined in one number sentence. In this case, it would be

17 ÷ 3 = 5 rem 2

This is read as “17, divide by 3, result 5 remainder 2.” or as “17 divided by 3 equals 5 remainder 2.” 3. They then repeat steps 1 and 2 for the other number sentences.

Activity 3 Unpacking the parcel (division) [Num 6.3/3] A continuation of Activity 2, for a small group. Its purpose is to remind children of

the two possible meanings of a number sentence for division.

Materials • Pencil and paper for each child.

What they do 1. Theyallwriteadivisionnumbersentenceontheirownpaper.Thefirstnumbershould not be greater than 20.

11 ÷ 4 = 2 rem 3

2. Thefirstchildshowshissentencetotheothers. 3. The next two on his left give the grouping and sharing meanings. In this case,

“Start with 11, make groups of 4, result 2 groups remainder 3” followed by “Start with 11, share among 4, 2 in each share (or, each gets 2), remainder 3.” Note that “. . . groups,” or “. . . in each share,” are important parts of the expand-ed meaning.

4. Steps 2 and 3 are repeated until all have had their sentences ‘unpacked.’

Num 6.3 Division as a mathematical operation (cont.)

191

Activity 4 Mr. Taylor’s Game [Num 6.3/4] This game for 2 players was invented by Mr. Stephen Taylor, of Dorridge Junior

School, and I am grateful to him for permission to include it here. Its purpose is to bring together addition, subtraction, multiplication, and division, in a simple game.

Materials • Number cards: 1 set 0 to 25, 3 sets 0 to 9.* • Game board, see Figure 13.* • Counters of a different colour for each player. * Provided in the photomasters

What they do 1. The object is to get 3 counters together in a line. They must be in the same row, column, or diagonal.

2. Thenumbercardsareshuffledandputfacedown,the0to25cardsinonepileand the 3 sets of 0 to 9 cards in another.

3. Thefirstplayerturnsoverthetopcardofeachpile.Hemaychoosetoadd,sub-tract, multiply, or divide the numbers shown. Division must, however, be exact.

4. He puts one of his counters on the square corresponding to the result of the op-eration chosen.

5. The other player turns over the next cards on the two piles and carries out steps 3 and 4.

6. Play continues until one player has 3 in a row. 7. Another round may now be played. The loser begins.

Activity 1 poses a problem, for children to solve by the activity of their own intel-ligence. When they have seen the connection between grouping and sharing, they have the higher order mathematical concept of division. The easiest path to seeing the correspondence is, I think, a physical one. If we have 15 objects to share among 3 persons, a natural way to do this is to begin by giving one object to each person. This takes 3 objects, a single ‘round,’ which we may think of as a group of 3. The next round may be thought of as another group of 3, and so on. Each round gives one object to each share, so the number of rounds is the number in each share. This verbal description by itself is harder to follow than a physical demonstra-tion accompanied by explanation. This I see as yet another demonstration of the advantage of combining Modes 1 and 2. Activity 2 provides a notation for these two aspects of division. Note that the firstnumberistheoperand,thatonwhichtheoperationisdone.Theoperationisthe division sign together with the second number, e.g., ÷ 3. Activity 3 is another example of ‘Unpacking the parcel.’ There is much less in this one than in the subtraction parcel, but is still a useful reminder of the two physical meanings combined in a single mathematical notation. Afterallthis,theydeserveagame.Mr.Taylor’sgamefitsinnicelyatthisstage.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Num 6.3 Division as a mathematical operation (cont.)

192

5 20 19 25 1

21 7 14 6 16

3 23 11 18 24

9 17 8 13 2

12 4 10 22 15

Figure 13 Mr. Taylor’s Game. © Stephen Taylor, 1981

Num 6.3 Division as a mathematical operation (cont.)

193

Num 6.4 ORgANIzINg INTO RECTANgLES

Concept A rectangular number.

Ability To recognize and construct rectangular numbers.

A rectangular number is one which can be represented as the number of dots in a rectangular array. Here are some examples.

How about this one?

The term ‘rectangle’ is currently used with two meanings: any shape which has a right angle at every corner, which includes both squares and oblongs; and ob-longs only. For myself, I agree with the teacher who said during a discussion of this point, “It makes no more sense to me to talk about squares and rectangles, than aboutgirlsandchildren.”SoIhopethatwemayagreeonthefirstmeaning,i.e.,that oblongs and squares are two kinds of rectangle. A rectangular number is of course the same as a composite number, but in this aspect it provides a very good link between multiplication and division, and an introduction to factoring.

Discussion of concept

• • • • •

15 • • • • •

• • • • •

• • • • 8

• • • •

• • •

9 • • •

• • •

194

Activity 1 Constructing rectangular numbers [Num 6.4/1] An activity for a small group, working in pairs. Its purpose is to build the concept of

rectangular numbers.

Materials For each pair: • 25 small counters. • Pencil and paper.

What they do 1. The activity is introduced along the following lines. Explain, “We think of these counters as dots which we can move around.”

2. Put out a rectangular array such as this one.

3. Ask, “What shape have we made?” (A rectangle.) 4. “How many counters?” (In this case, 12.) 5. “So we call 12 a rectangular number.” 6. Repeat, with other examples, until the children have grasped the concept. Note

that the rectangles must be solid arrays like the one illustrated. 7. Next, let the children work in pairs. Give 25 counters to each pair, and ask them

tofindalltherectangularnumbersupto25. 8. If any of them think that a pattern like this might be a rectangle, remind them that the counters represent dots, and ask: “Would you say this makes a rectangle?” 9. The question nearly always arises whether or not squares are to be included. I

suggest that you wait until someone raises this point, and then answer along the lines given in ‘Discussion of concept.’

10. Finally, let the children compare and check their lists with one another.

Num 6.4 Organizing into rectangles (cont.)

195

Activity 2 The rectangular numbers game [Num 6.4/2] A game for two. It consolidates the concept of a rectangular number in a predictive

situation. Children also discover prime numbers, though usually they do not yet know this name for them.

Materials • 25 counters. • Pencil and paper for scoring.

Rules of 1. Each player in turn gives the other a number of counters. the game 2. If the receiving player can make a rectangle with these, she scores a point. If

not, the other scores a point. 3. If when the receiving player has made a rectangle (and scored a point), the giv-

ing player can make a different rectangle with the same counters, she too scores a point. (E.g., 12, 16, 18).

4. The same number may not be used twice. To keep track of this, the numbers 1 to 25 are written at the bottom of the score sheet and crossed out as used.

5. The winner is the player who scores the most points.

InActivity1,thechildrenfirstformtheconceptofarectangularnumberfromphysical examples (Mode 1 building), and then construct further examples for themselves (Mode 3 building, Mode 1 testing). Activity 2 uses the new concept in a predictive situation (Mode 1 testing).

OBSERVE AND LISTEN REFLECT DISCUSS

Num 6.4 Organizing into rectangles (cont.)

Discussion of activities

The rectangular numbers game [Num 6.4/2]

196

Num 6.5 FACTORINg: COMPOSITE NUMBERS AND PRIME NUMBERS

Concepts (i) Factoring. (ii) Composite numbers and prime numbers. Abilities (i) To factor a given composite number. (ii) To distinguish between composite and prime numbers.

A composite number is one which can be written as the product of two (or more) numbers, other than itself and 1. E.g.,

15 = 3 x 5

22 = 11 x 2

Often this may be done in more than one way. E.g.,

24 = 2 x 12 24 = 3 x 8 24 = 4 x 6

To write a number as the product of other numbers is called factoring, and these other numbers are called factors of the original number. E.g., 3 and 8 are factors of 24. The process may often be continued, e.g.

140 = 14 x 10 = 2 x 7 x 2 x 5

which may be rearranged as

140 = 2 x 2 x 5 x 7

In this topic we shall only deal with 2 factors, since we are here seeing it largely as a preliminary to the use of multiplication facts for dividing larger numbers. 1 is a factor of every number. A number which has no factors other than 1 and itself is called prime. So the terms ‘rectangular number’ and ‘composite number’ are interchangeable; and numbers which are not rectangular numbers are prime.

Discussion of concepts

197

Activity 1 Factors bingo [Num 6.5/1]

A game for up to 6 players. Its purpose is to introduce the concept of factoring at a level where the calculations are still easy. It only requires knowledge of the multipli-cation tables up to 6 x 6.

Materials • 6 ‘factors’ bingo cards.* • Die 1-6 and shaker. • A non-permanent marking pen for each player. • Damp rag. * Provided in the photomasters. See example in step 5 below. These should be cov-

eredwithplasticfilm,sothattheycanbecleanedandre-used.

Rules of 1. Begin by explaining the meaning of ‘factor.’ the game 2. Each player has a Factors Bingo card and a non-permanent pen. 3. Players in turn, throw the die, and call out the number which comes up. This

number is available for use by everyone. 4. Theyfilltheircardsbywritingthegivennumberinanyspacewhereitisafactor

of the number on the left. This may only be done once for each throw. 5. A partly completed card might look like this, assuming that the numbers 2, 6, 4, 5, 3 have been thrown so far. (This player was not able to use 5.) 6. Thewinneristhefirsttofillhiscard. 7. Thisgameisnotquiteassimpleasitmightfirstappear.Ifaplayerwrites(say)

2 as one of the factors of 28, he is left with 14 as the other factor, and will not be abletofillhiscard.Asimilarproblemariseswith1.

Activity 2 Factors rummy [Num 6.5/2]

A game for up to 6 players. Its purpose is to develop mental agility in factoring.

Materials • Product pack: one card each of the numbers 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 36.*

• Factors pack: six cards each of the numbers 2, 3, 4, 5, 6, 7, 8, 9.* * Provided in the photomasters. These two packs should be double headed (to facili-

tate holding a ‘hand.’) and of different colours.

1. The product pack is put face down on the table. The dealer deals 5 cards from the factors pack to each player, and puts the rest of the factors pack face down on the table.

2. He then turns over the top card of each pack, and puts it face up beside the pile. 3. The object is to get rid of one’s cards by putting down pairs of cards which,

when multiplied together, make the product shown on the face-up product card.

4 = 2 x 6 = x24 = 6 x 418 = 3 x12 = x

Num 6.5 Factoring: composite numbers and prime numbers (cont.)

Rules of the game

198

4. Players begin by looking at their cards, and putting down any pairs they can. 5. Thefirstplayerthenpicksupafactorcardfromeitherthefaceuportheface

down pile, whichever he prefers. If he now has a pair, he puts it down. Finally he discards one of his cards onto the face up pile.

6. In turn the other players pick up, put down a pair if they can, discard. 7. At the end of each round, the dealer turns over the top product card from the face

down pile, and puts it face up beside (not on top of) the product card already showing. In this way, the number of possible products increases for every new round.

8. After step 7, players put down any pairs now made possible, before the next round is played.

9. Steps 5 to 8 are repeated, until a player puts down all his cards. 10. Play then stops, and each player scores the total of the cards still in his hand.

The player who went ‘out’ scores zero. 11. In this game, it is possible for two players to go ‘out’ simultaneously when a

new product card is turned. This does not matter: they both score zero. 12. The scores are recorded, and another round may then be played. 13. The winner is the player with the lowest total score.

Activity 3 Alias prime [Num 6.5/3]

A game for up to 6 players. Its purpose is to introduce children to the difference between composite and prime numbers, and give them practice in distinguishing between these two kinds of number.

Materials • Three counters for each player.

1. Begin by explaining the meanings of ‘composite number’ and ‘prime.’ These concepts have been well prepared in Activities 1 and 2, and in topic 4. 2. Explain that ‘alias’ means ‘another name for,’ often used to hide someone’s iden-

tity. In this game, all prime numbers use the alias ‘Prime’ instead of their usual name.

3. Start by having the players say in turn “Eight,” “Nine,” “Ten,” . . . round the table.

4. The game now begins. They say the numbers around the table as before, but when it is a player’s turn to say any prime number, its usual name must not be said but “Prime” is said, instead.

5. The next player must remember the number which wasn’t spoken, and say the next one. Thus the game would begin (assuming no mistake):

“Eight,” “Nine,” “Ten,” “Prime,” “Twelve,” “Prime,” “Fourteen,” “Fifteen,” “Sixteen,” “Prime,” “Eighteen,” and so on.

6. Any player who makes a mistake loses a life – i.e., one of his counters. Failing to say “Prime,” or saying the wrong composite number, are both mistakes.

7. When a player has lost all his lives he is out of the game, and acts as an umpire. 8. The winner is the last player to be left in the game.

Note When the players are experienced, they may begin counting at “One.” This gives rather a lot of primes for beginners!

Num 6.5 Factoring: composite numbers and prime numbers (cont.)

Rules of the game

199

Activity 4 The sieve of Eratosthenes [Num 6.5/4] An activity for children to play individually, or in small groups. Its purpose is to

teach them a classical piece of mathematics for identifying primes.

Materials • Number squares 1-100, see Figure 14.* • Pencil for each child. * One each, or shared. Provided in the photomasters.

What they do 1. In the previous game, when players are good enough to reach largish numbers, there is sometimes doubt whether a number is prime or not. Explain that this is a method which gets rid of all the composite numbers and leaves only primes.

2. Starting with 2 (the smallest prime other than 1), they cross out every multiple of 2 except 2 itself.

3. The next prime is 3, so they cross out every multiple of 3 except 3 itself. 4. Likewise for each successive prime. 5. The numbers remaining when the process is complete are all primes. 6. After a while, the children will notice that no new numbers are being crossed

out. With numbers up to 100, the process is complete when all the multiples of 7 have been crossed out.

7. The intriguing question is, Why? When we come to cross out the multiples of 11,13,etc.,whydowefindthatthesehavealreadybeendone?Whilesteps1to6 are straightforward, this requires careful thought.

8. A simpler question is, “Why do we only cross out multiples of primes?” 9. This method is quite general, and may be extended beyond 100 if desired.

In this topic we are mainly concerned with introducing the concept of factoring as a preparation for its use in division, so that children will later be able to use their knowledge of the multiplication tables for dividing. This saves learning a whole new body of facts. For this reason the numbers involved are kept small. Activi-ties 3 and 4 develop an offshoot of factoring, namely prime numbers. We do not develop the study of primes any further than this, but they have long intrigued mathematicians, and children should at least know what they are. Using their knowledge of multiplication results to arrive at division results is, of course, a strong example of Mode 3 schema building. Doing things backwards is usually harder than forwards, and it certainly is in this case. So division, both conceptually and as a skill, is prepared and developed carefully over a number of topics.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Num 6.5 Factoring: composite numbers and prime numbers (cont.)

200

Figure 14 The sieve of Eratosthenes

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Num 6.5 Factoring: composite numbers and prime numbers (cont.)

201

Num 6.6 RELATION BETwEEN MULTIPLICATION AND DIVISION

Concept The relation between multiplication and division.

Ability To express the same relationship in a variety of different ways: as multiplying, grouping, sharing, dividing.

The new concept to be formed in this topic is an awareness of how all of the fol-lowing are connected.

grouping

multiplying dividing

sharing

This is also a convenient place to introduce vertical notation for division, in preparation for dividing larger numbers in the next topic.

Activity 1 Parcels within parcels [Num 6.6/1] A game for up to 6 children. Its purpose is to make explicit the relationships be-

tween multiplication and the various aspects of division.

Materials • 6 or more parcel cards.* • Pencil and paper for each child. • Bowl of counters. * Provided in the photomasters

Rules of Stage (a) the game 1. Theparcelcardsareshuffledandputfacedown. 2. Thefirstplayerturnsoverthetopparcelcard. 3. Shethenstartsto“unpacktheparcel”byfirstwritinganumbersentencewhich

she shows to the others, and then reading it aloud in as many ways as she can. (See the example parcel card at the top of the page, overleaf.)

4. Iftheothersagree,theysay“Agree.”Thefirstplayertakesacounterforeachagreed statement.

5. Iftheydisagree,thefirstplayerhastojustifyhisstatementbyreferencetotheparcel card. E.g., for 3(7) = 21, she would point out that there are 3 rows of 7, making 21.

6. Steps 3,4,5 are repeated by the other players in turn until the parcel has been fully unpacked.

7. The next player turns over the top card, and steps 1-6 are repeated. 8. The winner is the player with the most counters at the end.

Discussion of concepts

202

Example Parcel card turned over:

Contents A. Writes: 3(7) = 21 Reads: “3 sets of 7 make a set of 21” or “3 sevens are/make/equal 21” (1 point) B. Likewise for 7(3). (1 point) C. Writes: 3 ∗ 7 = 21. Reads: “3 star 7 equals 21.” “3 sevens are 21.” “7 threes are 21.” ( 3 points) D. Writes: 21 ÷ 3 = 7 Reads: “21 divided by 3 equals 7” “21, make groups of 3, result 7 groups.” “21, share between 3, result 7 in each share.” (3 points) E. Writes: 21 > 7 Reads: “21, make groups of 3, result 7 groups.” (1 point) F. Writes: 21 > 7 Reads: “21, share between 3, result 7 each.” (1 point.)

G,H,I. Likewise for 21 ÷ 7 = 3.

As may be seen from this example, it is quite a sophisticated game. C, D and G are clearlytheonestochoosefirst.Buttherearestillpointstobecollectedwhenthesehave been used.

Stage (b) 1. When they have a good grasp of the foregoing, introduce the following notation.

Explain that this is another way of writing a division sentence, which is needed for larger numbers.

This is read as: “How many 5’s in 15? Answer, 3.” 2. It should notbereadas“fiveintofifteen...”,asonesometimeshears.Weare

certainly not dividing 5 into 15 shares. 3. Continue as in Stage (a), including this notation.

Num 6.6 Relation between multiplication and division (cont.)

make groups of 3

share between 3

3 5 ) 15

203

Stage (a) is conceptually quite complex. This is a necessary consequence of the great variety of information which can be extracted from one ‘parcel.’ All these waysofthinkingareneededfordifferentpurposes;anditismuchmoreefficient,and economical of memory, to learn them all as different ways of thinking about the same number-relationship. This relationship is represented much more clearly by the rectangular array than it can be by any notation, each of which shows only a particular aspect. All these aspects are concepts already learned. What the chil-dren are now doing is becoming more aware of their interrelationships. Stage (b) is simpler, since it only involves learning a new notation for what they already know. Note that in both stages of the activity, the main links which are being strength-ened are between notations and their various meanings, not between one notation and another. Their various meanings are all implicit in the rectangular arrays – the parcel cards.

OBSERVE AND LISTEN REFLECT DISCUSS

Num 6.6 Relation between multiplication and division (cont.)

Discussion of activity

Parcels within parcels [Num 6.6/1]

204

Num 6.7 USINg MULTIPLICATION RESULTS FOR DIVISION

Concept The use of multiplication results in reverse, to obtain division results.

Ability To arrive at a division result mentally by recalling an appropriate multiplication result.

The close relationship between the two statements

has already been learned, using small numbers and visual representations. In this topic the concept is expanded to include all products up to and including 10 x 10, and made independent of visual representations.

Activity 1 A new use for the multiplication square [Num 6.7/1]

An activity for children to do in pairs. Its purpose is to develop the ability described above.

Materials • Multiplication square for each child.* • 1 to 9 die. • Pencil and paper for each child. * They should still have the multiplication squares they made for themselves in Num 5.6/3. A photomaster is also provided in Volume 2a.

What they do 1. Player A has his multiplication square face up. Player B has his square face down.

2. Player A throws the die. Suppose it shows 7. He therefore takes the row starting with 7, and writes

7 3. He then chooses any number in this row, say 35. He writes

7 ) 35 and passes the paper to Player B. 4. PlayerBfirstreadsthisaloud,as“Howmanysevensin35?” 5. Next, if he can, he recalls from his multiplication tables that 5 sevens are 35. He

says this aloud, and completes the written division. 5 7 ) 35 6. Ifhecannotdothis,heturnsoverhismultiplicationtable,findstherowstarting

7, and then locates 35 in this row. He uses this to say aloud, “5 sevens are 35,” and then continues as in step 6.

7. In either case, player A checks B’s result and says “Agree.” 8. The players then interchange roles, and steps 2 to 6 are repeated. 9. Players should be encouraged to give each other practice with all the products in

a row, and not just the harder ones. For this reason, no scoring system is includ-ed in this activity, which might cause players to give each other only the more difficultexamples.

Discussion of concept

“3fivesare15” and “Howmanyfivesin15?Answer3.”

205

Activity 2 Quotients and Remainders [Num 6.7/2] An activity for up to 4 children. Its purpose is to give practice in division, and to

introduce the terms ‘quotient’ and ‘remainder’ using easy numbers.

Materials • Q and R board,see Figure 15.* • Start cards 9 to 30. • Die 1-6. • For each child, 5 counters of 1 colour. • Pencil and paper for each child (optional). * Provided in the photomasters

1. Thestartcardsareshuffledandputfacedown. 2. Thefirstplayerthrowsthedie,andthenturnsoverthetopstartcard.Thisgives

a number to divide by, and a starting number; e.g., 3 and 23. 7 rem 2 3 ) 23 3. He says, “Quotient 7, remainder 2,” so that the others can check. 4. He may then place one of his counters on a 7 and one on a 2, the aim being to

get 3 counters in a row. This row may be across, down, or diagonal. 5. The next player repeats steps 2 to 4, and so on in turn until either a player has

won by getting 3 in a row, or this has become impossible. 6. A player who throws 1 may have another turn. If the quotient is a number not on

the board, only the remainder is used. 7. A new game may then be played. 8. (Optional) Pencil and paper may be used for written calculation.

Note This board has been designed to give a quick result. For a longer game, play contin-ues until each player has put down all his counters. For the last counter, he chooses either the quotient or the remainder. The winner is then the player with the longest row.

Num 6.7 Using multiplication results for division (cont.)

Rules of the game

Quotients and Remainders [Num 6.7/2]

206

Figure 15 Q and R board

Num 6.7 Using multiplication results for division (cont.)

0 6 2 5 1 7 0

3 1 8 2 4 3 1

9 8 4 6 5 4 10

3 1 5 0 10 2 4

9 7 2 5 1 3 10

4 3 5 4 3 5 2

0 9 7 2 6 8 0

207

Activity 3 VillagePostOffice[Num 6.7/3] A game for 2 to 6 children. Its purpose is to use the relationship between division

and multiplication in situations where division involves a remainder.

Materials • Multiplication square.* • Notice.*† • Stamp cards, 3 to 10.* Parcel cards.** • Pencil and paper for each child. * Provided in the photomasters † See illustration in step 3 below **As many as you like. I suggest that children begin with an easy pack, with num-

bers up to 30¢; and progress to a harder pack, up to 70¢. Suitable numbers are:

Easy 11, 14, 17, 19, 23, 25, 26, 28, 29, 30. Hard 31, 33, 35, 36, 37, 40, 41, 43, 46, 53, 55, 57, 62, 65, 67, 69.

With the harder set of parcel cards, only stamp cards 7-10 should be used.

Rules of For 2 children the game 1. One acts as postal clerk, the other as customer. 2. Thepostalclerkshufflesthestampcards,thecustomershufflestheparcelcards,

and these are then put face down in separate piles. 3. The postal clerk opens the shop by turning over the top stamp card and putting it

in the space in the notice. It now reads (e.g.),

4. The customer then takes the top parcel card and turns it over. Suppose it reads 26¢. He calculates 26 ÷ 6 = 4 rem 2 and should therefore ask for 4 six-cent and 2 one-cent stamps. Note that the postal clerk likes customers to ask for the least number of stamps which gives the required total. (He doesn’t take kindly to requests for 26 one-cent stamps.)

5. The postal clerk checks: 4(6) + 2 = 26. 6. If correct, he says “Correct” and accepts the parcel. If the results do not agree,

the multiplication square may be used to discover where their mistake lies. 7. The next parcel card is then taken by the customer, and steps 4 to 6 are repeated. For up to 6 children In addition to the postal clerk, there are now 1 or 2 assistants. The remaining chil-

dren are customers, and are served by whichever clerk is free. Otherwise the game is played as above.

Num 6.7 Using multiplication results for division (cont.)

208

In the earlier topics, learning has been directed towards the establishment of the concept of division in its two aspects, grouping and sharing; and with relating ittomultiplication.Inthepresenttopic,thepurposeistodevelopfluencyincalculation. The basis of this is the use of already-known multiplication facts for division (Activity 1). The notation which children will need for dividing larger numbers is also used here. Activity 2 gives further practice at these in the form of a game, still using fairly small numbers. Activity 3 is important for linking these calculation skills back to the concepts involved. The terms ‘quotient’ and ‘remainder’ are useful ones for later work. I leave it for you to decide whether to include ‘divisor’ and ‘dividend’ also.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Num 6.7 Using multiplication results for division (cont.)

Village Post Office [Num 6.7/3]

209

Num 6.8 DIVIDINg LARgER NUMBERS

Concept The concept of division, as already learned.

Ability Todividea2,3,or4-digitnumberbyasingledigitnumber,firstwithoutregroupingand then with regrouping.

The concept of division has already been learned. What we are here concerned with is extending the ability to larger numbers. The method for doing this uses non-canonical form, which has already been used in addition, subtraction, and multiplication.

Activity 1 “i’m thinking in hundreds . . . .” [Num 6.8/1] Anactivityforchildreningroupsof3.Itspurposeistoteachthemthefirststepsin

the ability described above.

Materials For each group: • HTOnes division board, laminated.* • Non-permanent pen and wiper. • Examples card.** * See illustration in step 2 below. A division board is provided in the photomasters,

from which expendable sheets can be reproduced. ** These are the examples:

684 ÷ 2 936 ÷ 3 848 ÷ 4 408 ÷ 2 630 ÷ 3 880 ÷ 4 690 ÷ 3 906 ÷ 3 208 ÷ 2 550 ÷ 5

What they do 1. The children need to sit facing the same way. 2. ThefirstoftheexamplesiscopiedinthelefthandspaceoftheHTOnesdivision

board by the child sitting in the middle, using the notation shown.

She reads this aloud: “How many 2’s in 684?”, and passes the board and non-permanent pen to the child on her left.

3. The left hand child says, “I’m thinking in hundreds,” and writes the hundreds figureinthehundredscolumn.

Discussion of concepts

210

4. Next she writes, reading aloud as she does so: “How many 2’s in 6 hundreds? Answer, 3 hundred.” (N.B. 3 hundred, not 3.)

5. Steps3and4arerepeatedforthetensfigurebythemiddlechild.Shebeginsby

saying, “I’m thinking in tens.” 6. Steps3and4arerepeatedfortheonesfigurebytherighthandchild.Shebe-

gins, “I’m thinking in ones.” 7. The lower part of the board should now look like this.

8. The board is then passed back to the middle child, who completes the number sentence on the left.

She reads it aloud. “How many two’s in six hundred eighty four? Answer, three hundred forty-two.”

9. The board is wiped clean, and steps 2 to 8 are repeated with the next example on the card.

10. The order in which the children sit should be varied.

Activity 2 “i’ll take over your remainder.” [Num 6.8/2] A continuation of Activity 1, for children in groups of 3. Its purpose is to teach them

how to deal with remainders in each column.

Materials For each group: • HTOnes division board, laminated.* • Non-permanent pen and wiper.* • Examples card.** * The same as for Activity 1.

Num 6.8 Dividing larger numbers (cont.)

211

Num 6.8 Dividing larger numbers (cont.)

** Here are 10 examples. Others may be taken from a suitable textbook. You may wishtogradethedifficultymoreslowly.

743 ÷ 3 851 ÷ 3 783 ÷ 4 680 ÷ 5 705 ÷ 6

241 ÷ 7 354 ÷ 8 559 ÷ 9 303 ÷ 4 600 ÷ 7

What they do 1. ThefirstexampleiscopiedinthelefthandspaceoftheHTOnesdivisionboardby the child sitting in the middle, and read aloud, as in Activity 1. The board is then passed to the child on the left.

2. Theleft-handchildwritesthehundredsfigureinthehundredscolumn.

3. Next she writes, reading aloud as she does so: “How many 3’s in 7 hundreds? Answer, 2 hundred, remainder 1 hundred.”

4. The question is now, what to do with the remainder? 5. Themiddlechildwritesthetensfigureinthetenscolumn.

212

6. Next, she says, “I’ll take over your remainder. 1 hundred is 10 tens, so that makes 14 tens.” The remainder in the hundreds column is crossed out.

7. He writes, and reads aloud: “How many 3’s in 14 tens? Answer, 4 tens, remainder 2 tens.”

8. The right hand child deals similarly with the ones. He takes over the remainder of 2 tens, which are 20 ones, making 23 ones altogether.

9. The board is then passed back to the middle child, who completes the number sentence on the left.

She reads it aloud. “How many threes in seven hundred and forty three? An-swer, two hundred and forty seven, remainder 2.

10. The board is wiped clean, and steps 1 to 9 are repeated with the next example. The order in which the children sit should be varied.

11. When children have had plenty of experience of the above, transition to the con-densed notation is straightforward. To begin with, the same calculation should be rewritten in the condensed notation to show the correspondence. In the above example, this would be:

Num 6.8 Dividing larger numbers (cont.)

rem 1

rem 1

r 2

213

This condensed notation is a good one, provided that its meaning is well under-stood. Now that calculators are widely available, I attach little importance to children’s acquiring a high degree of skill in this kind of computation; but it is good to understand the principle.

Activity 3 Q and R ladders [Num 6.8/3] Thisgameissuitablefor4or6children,playinginteamsof2.Itspurposeisfirst,to

expand the abilities developed in Activities 1 and 2 to larger numbers; and second, to developfluencyinthesecalculations.

Materials • Game board.* • Markers: a pair of one colour for each team of 2. • Start cards 50 to 100.* • Die 5 to 10.** • Pencil and paper for each pair, or single player. * See the photomasters in Volume 2a. The game board consists of two ladders, one

for quotients and one for remainders. ** Make your own from 1 cm cubes.

Rules of 1. Thestartcardsareshuffledandputfacedown. the game 2. Each team puts one of its markers on the zero rung in each ladder. These repre-

sent climbers. 3. Thefirstpairbeginsasfollows:oneplayerturnsoverthetopstartcard,andthe

other throws the die. 4. They then co-operate in dividing the number on the start card, by the number on

the die, using the method learned in Activity 2. This gives them a quotient and a remainder.

5. These two numbers show how many rungs their two pieces may climb, each up its respective ladder.

6. The other teams in turn repeat steps 3 to 5. 7. The winning team is that which gets both of its two climbers to the top of the

ladder, or beyond. The exact number is not required in this game.

Note This game usually takes 6, 7, or 8 rounds; occasionally, fewer. It has been designed sothatthelikelihoodofbeingfirstatthetopisapproximatelyequalforclimbersonthe two ladders. There is also about a 1 in 5 chance of both getting there at the same time.

Num 6.8 Dividing larger numbers (cont.)

214

Activity 4 Cargo Airships [Num 6.8/4] A game for up to 4 players. Its purpose is to give them plenty of practice in division

of numbers up to 1000, by numbers from 3 to 9.

Materials • Cards representing airports, one for each player.* • 7 cards representing airships.* • Cards representing loads.*† • Loading slips.* • Calculators. • Pencil and paper for each player. • A paper clip for each airship. * Photomasters provided in SAIL Volume 2a. † A suitable set of numbers is 325, 108, 473, 235, 90, 156, 209, 392, 267, 340,

440, 60, 83, 116, 180, in each case followed by ‘tonnes.’ (See photomasters.)

Rules of 1. This game is like One Tonne Van Drivers, but harder since it involves dividing. the game 2. Each cargo airship can carry from 600 to 1000 tonnes, according to size. They

also vary from 3 to 9 in their number of compartments, and the loads must be equally shared between the compartments to keep the airship level. This is the taskofeveryloadingofficer.

3. Alltheplayersexceptoneactasloadingofficersatdistantairports.Theremain-ingplayeristhecheckingofficeratthehomeairport.Hehasacalculator.

4. Eachloadingofficerhasacardrepresentinghisairport,andanairship.Theair-ships are assigned randomly.

5. Tobeginthegame,theloadcardsareshuffledandputfacedowninthemiddleof the table.

6. The loads arrive by road at the distant airport. To represent this in the game, the firstloadingofficertakesthetoploadcard.

7. This must be shared equally among whatever number of compartments the air-crafthas.Sotheloadingofficercalculatesthecorrectpart-loads,equalforeachcompartment. These will be carefully weighed out on the scale before loading. E.g., if he has a 6 compartment airship, and a 473 tonne load arrives, he would instruct his loaders to make part-loads of 78 tonnes each, leaving a remainder of 5 tonnes. He writes this on a loading slip, as shown in step 10.

8. Sincetheremaindersaresmallcomparedwiththepartloads,thecargoofficeronthe airship uses his own discretion where to put these.

9. Steps6and7arerepeatedinturnbytheotherloadingofficers,andthiscontin-ues until one of them receives a load which would make his airship overloaded. He must then send that airship off to the home port, together with a loading slip which might look like this.

10. Airship Sky King Max. load 900 tonnes, 6 compartments

Load 6 part-loads, each remainder

473 78 5 108 18 0 235 39 1 Totals 816 135 6

Num 6.8 Dividing larger numbers (cont.)

215

Figure 16 Cargo airships

Discussion of activities

Num 6.8 Dividing larger numbers (cont.)

11. Theotherloadingofficerssendhometheirairshipswhenthesecantakenomore. 12. Aseachairshiparrivesbackatthehomeairport,thecheckingofficerchecks

by calculator all the relevant calculations, including part-loads and remainders. There is good opportunity for discussion about how best to do this. The unused capacity is also calculated, in the above example 84 tonnes (900 - 816).

13. Miscalculations are penalized by adding 100 tonnes to the unused capacity for each error in a part load, and 5 tonnes for each error in a remainder.

14. Thebestloadingofficeristheonewiththeleastunusedcapacity.

Activities 1 and 2 are for extending children’s calculating skills to larger dividends, stillwithsinglefiguredivisors.Thesewouldconventionallybedonebychildrenworking singly, using pencil and paper and very likely the conventional notation. The thinking behind the present approach, in which three children work together using a shared calculation board, is as follows. By making it a social activity, it be-comes more interesting and introduces discussion. Each child sees and checks the accuracy of the calculations of the others. In Activity 1, by having 3 children each say aloud, “I’m thinking in . . . ,” they emphasize the changes in thinking between columns. This is important preparation for passing down the remainders in Activ-ity 2. Both the headed columns notation, and saying aloud what they are doing, are being used to retain awareness of what is happening at each stage, so that when this method is eventually replaced by the conventional (and highly condensed) no-tation, children will, we hope, have established their understanding of the thinking processes which it represents.

216

3 2 ) 6

Num 6.8 Dividing larger numbers (cont.)

I have introduced the method with HTOnes from the start, rather than TOnes, because it is no harder and gives two examples of passing on the remainder each time rather than one. But if you prefer to begin with TOnes, by all means do so. The method of calculating used in this topic is a hybrid. It is much easier to think of

as “How many 2’s in 6? Answer 3” than as “Start with 6, share between 2, 3 in eachshare”;soitispresentedthefirstway.Butifwewereworkingwithphysicalmaterials,wewoulddoitbysharingfirstthehundreds,thenthetens,thenthesin-gles. (The reason for not using physical materials here is that very large quantities would be needed. With a divisor (say) 8, there could be a remainder of 7 hundred-cubes to be exchanged for 70 ten-rods.) And as already noted, “2 into 6 goes 3 times” does not describe what is being done – it is inaccurate and confusing. Myjustificationforthishybridisthatoncewehaveestablishedgroupingandsharing as mathematically equivalent (topic 3), they become interchangeable for purposes of calculation. (We think in the same way when we know that 3 tens and 30 ones are equivalent, and therefore interchangeable for the purposes of calcula-tion.) So when we are concerned with the mathematical operation of division, we arejustifiedinusingwhicheverofthetwovarietiesbestservesourthinking. Activities 3 and 4 are games which give plenty of practice in these new meth-ods. Activity 3 (Q and R ladders) involves only division of tens and ones; Activity 4, division of hundreds, tens and ones.

OBSERVE AND LISTEN REFLECT DISCUSS

Cargo Airships [Num 6.8/4]

217

Num 6.9 DIVISION By CALCULATOR

Concept The calculator as a tool by which we can divide, whatever the numbers, easily and quickly.

Abilities (i) To use a calculator correctly. (ii) To interpret the results. (iii) To deal appropriately with any decimal fractions which appear in the results.

The widespread availability of inexpensive calculators has a number of advantages for school mathematics. By reducing the amount of time which has to be spent on calculating, time is released which can be spent more usefully in other ways. They also allow the use of numbers which arise naturally: there is no longer any need to useartificiallysimplifiednumberswhich‘workoutnicely.’ In general, I think that children should continue to learn the skills of addition, subtraction, multiplication, and division. This will be obvious from the amount of space given to these in the networks. But I cannot see any good reason why chil-dren should continue to learn long division. No new mathematical understanding results,andcomparedwiththetimeandeffortinvolvedthebenefitsarenegligible. What I do see as important is that calculators should be used with understand-ing, and this is the aim of this topic. Decimal fractions are dealt with in the Fractions network, which is where they belong. Rounding also comes there, with the help of the a number line. Both of these topics should be done before the present one.

Activity 1 number targets: division by calculator [Num 6.9/1] A game for up to 6 children, though a smaller number is better. Its purpose is to

develop further the relationships between dividend, divisor, and quotient, with more difficultnumbers.Theuseofacalculatorfreesthemindfromthedistractionoflabo-rious calculations, and allows full attention to be given to these relationships. This is also a convenient time to introduce the terms ‘dividend’ and ‘divisor,’ if this has not been done already.

Materials • Calculator. • Three 0-9 dice. • Paper and pencil.

Rules of 1. The three dice are thrown to give a 3-digit number. This is written on the paper the game as the dividend. 2. Two dice are then thrown to give a 2-digit number. This is written as the target.

E.g.

Discussion of concepts

Dividend Target 747 64

218

3. Theaimistofindanumberwhich,whenusedasdivisor,givesthetargetnumberas quotient when the calculator result is rounded to the nearest whole number. Each player in turn writes his attempt, and passes paper, pencil, and calculator to the next player. Example:

747 ÷ 12 = 62.25 (Goodforafirsttrial.) 747 ÷ 13 = 57.461539 (This player changed the divisor in the wrong direction). 747 ÷ 11 = 67.909091 (This player learned from the other’s mistake). 747 ÷ 11.5 = 64.956522 (Nearly there, but this rounds to 65). 747 ÷ 11.6 = 64.396552 (He wins this round.)

4. Another round may now be played, repeating steps 1 to 3. The winner of the last roundstarts,sincethefirstattemptisthemostdifficult.

Activity1usesacalculatortotestpredictions.Itisquiteadifficultactivity,requir-ing children to take into account both the direction in which the divisor needs to be changed from the last trial, and also what is a likely amount. This seems to me a much better direction in which to develop their understanding than learning the laborious, time-consuming, and no longer necessary procedure of long division.

OBSERVE AND LISTEN REFLECT DISCUSS

Dividend Target 747 64

Num 6.9 Division by calculator (cont.)

Discussion of activities

Number targets: division by calculator [Num 6.9/1]

219

[Num 7] Fractions Double operations numbers which represent these Fractions as quotients

Num 7.1 MakiNg equal parts

Concepts (i) The whole of an object. (ii) Part of an object. (iii) Equal parts and their names.

Abilities Tomake,name,andrecognizewholes,halves,third-parts,fourth-parts,fifth-parts,etc., of a variety of objects.

Manychildrenhavedifficultywithfractions,andIthinkthereareseveralcauseswhich continue to bring this about. (i) Fractions aredifficult.Workwithfractionsisbeguntooearly,andtakentoofar for children of elementary school age. (ii) The same notation is used with three distinct meanings. For example, 2

3can mean a fraction, or a fractional number, or a quotient.(iii) The authors of many text books for children appear to confuse these three meanings,andtheypassontheirconfusiontothechildren.Itisratherlikeconfus-ing the different physical embodiments of 7 - 3: take-away, comparison, comple-ment. But in this case the confusion is at a more abstract level. The distinction isnotaneasyone,andafulldiscussionofthisIwouldconsiderasatleasthighschool mathematics. InthepresentnetworkIhavetriedtopresentonlythetruth,butforthereasonsabovenotthewholetruth.Webeginasusualbyintroducingtheconceptinsev-eral different physical embodiments. The terms ‘third-part,’ ‘fourth-part,’ etc., are used to distinguish fractions from the ordinal numbers third, fourth, etc.; and also as reminders that we are talking about parts of something. This is implicit in the word ‘part,’ but to start with we need to say so explicitly. Note that in this topic we arenotyettalkingaboutfractions,butaboutwholesandparts.Thisisjustthefirststage of the concept.

activity 1 Making equal parts [Num 7.1/1] Anactivityforupto6children.Itspurposeistomakeastartwiththeconceptofa

fraction, using two different physical embodiments.

Materials • SAR boards 1, 2, and 3 for ‘Making equal parts.’ * • Plasticine. • Blunt knives. • Cutting boards. * See Figure 17. A full-size version of board 1, together with boards 2 and 3, will be

found in the photomasters. For board 2, it is helpful also to have a cardboard tem-plate the size and shape of a fruit bar, which children can cut around.

Discussion of concepts

220

Num 7.1 Making equal parts (cont.)

Figure 17 SAR board 1. Making equal parts

start action rEsULt naME

A Plasticine sausage.

Leave this one (Put it The whole

as it is. here.) of a sausage.

A Plasticine sausage.

Make (Put them These are

2 here.) halves

equal parts. of a sausage.

A Plasticine sausage.

Make (Put them These are

3 here.) third-parts

equal parts. of a sausage.

A Plasticine sausage.

Make (Put them These are

4 here.) fourth-parts

equal parts. of a sausage.

Also called quarters.

A Plasticine sausage.

Make (Put them These are

5 here.) fifth-parts

equal parts. of a sausage.

221

Num 7.1 Making equal parts (cont.)

What they do (apportioned between the children) 1. Six equal-sized round Plasticine ‘sausages’ are made, by rolling 6 equal amounts

ofPlasticineandtrimmingtheendstofitthesausageshapesontheleftofboard1. One sausage is put into each outline.

2. The children do the actions described on the board. The lines of division may be markedlightlybeforecutting.Inthisway,trialscanbemadeandcorrectedbysmoothing out the marks.

3. After cutting, the separated parts are put in the RESULT column next to their descriptions.

4. Steps 1 to 3 are repeated using SAR board 2 and fresh Plasticine. Board 1 shouldifpossibleremainonview.Withboard2,avarietyofdivisionlinesareeasily found. E.g., fourth-parts:

At this stage accept any correct results. These possibilities will be explored fur-ther in the next activity.

5. Steps1to3arenowrepeatedusingSARboard3.Ifpossible,freshPlasticineshould be used, the other two boards remaining on view. The lines of division should be radial, as shown below.

activity 2 same kind, different shapes [Num 7.1/2] Anactivityforupto6children.Itspurposeistodeveloptheideathatpartsofthe

samekindmaynotlookalike.InActivity1,thisarosefromtheuseofdifferentob-jects. Here we see that this can be so even with the same object.

Materials • SAR boards: halves, fourth-parts, third-parts, see Figure 18.* • Plasticine. • Blunt knives. • Cutting boards. *Figure18illustratesthefirstoftheSARboardsforthisactivity;theothersarein

the photomasters.

222

Figure 18 SAR board 1. Same kind, different shapes. Halves

Num 7.1 Making equal parts (cont.)

in a different way.

in a different way.Therearethreesimpleways.Lateryoumaybeabletofindothers.

What they do 1. Begin with the halves board. This is used in the same way as the SAR board for Activity 1. The three straightforward ways are:

2. Next, they use the third-parts board. This offers only two straightforward ways.

3. Next, they use the fourth-parts board. There are six ways of doing this which are fairlyeasytofind.

start action rEsULt naME A fruit bar.

Make (Put them These are

2 here.) halves

equal parts. of a fruit bar.

Make (Put them These are also

2 here.) halves

equal parts of a fruit bar.

Make (Put them These are also

2 here.) halves

equal parts. of a fruit bar.

223

4. Finally,theymayliketoreturntothehalvesboard,andtrytofindmoreways.Hereare2more,whichcanbevariedindefinitely.

activity 3 Parts and bits [Num 7.1/3] Ateacher-leddiscussionforupto6children.Itspurposeistoemphasizethatwhen

we talk about third-parts, fourth-parts, etc., we mean equal parts. To make this clear, we use the word ‘bits’ when these are unequal.

Materials • Plasticine. • Blunt knives. • Cutting boards.

Suggested 1. Make 2 Plasticine ‘sausages’ of equal size. Cut one into 3 equal parts, the sequence for other into three bits. Ask whether both of these have been cut into third-parts. discussion 2. Ifnecessary,explainthedifference,asdescribedabove. 3. Have the children make some more sausages, all of the same size, and cut some

intohalves,fourth-parts,fifth-parts,andsomeintotwo,four,fivebits. 4. These are put centrally together with the parts and bits made in step 1. 5. The children then ask each other for parts or bits. E.g., “Sally, please give me a

fourth-part of a sausage”; or “Mark, please give me a bit of a sausage.” 6. The others say whether they agree. 7. Steps 5 and 6 are repeated as necessary.

activity 4 sorting parts [Num 7.1/4] Anactivityforupto5children.Itspurposeistoconsolidatetheconceptsformedin

Activities 1 and 2, moving on to a pictorial representation.

Materials • Parts pack of cards.* • 5 name cards.** • 5 set loops. * Some of these are illustrated in Figure 19. The complete pack is in the photomas-

ters. **ThesearemarkedWHOLES,HALVES,THIRD-PARTS,FOURTH-PARTS,

FIFTH-PARTS.

What they do 1. Begin by looking at some of the cards together. Explain that these represent the objects which they made from Plasticine, in the last activity – sausages, fruit bars, cookies; and some new ones. They also represent the parts into which the objectshavebeencut,e.g.,third-parts,fourth-parts,halves,fifth-parts.Somehave not been cut: these are wholes.

Num 7.1 Making equal parts (cont.)

224

Figure 19 Cards for Sorting parts and Match and mix: parts

Num 7.1 Making equal parts (cont.)

225

2. Thepartspackisthenshuffledandspreadoutfaceupwardsonthetable. 3. The name cards are put face down and each child takes one. 4. Each child puts in front of her a set loop with the name card she has taken inside,

face up. 5. They all then collect cards of one kind, according to the name card they have

taken. 6. They check each other’s sets, and discuss, if necessary. 7. Iftherearefewerthan6children,theymaythenworktogethertosortthere-

maining cards. 8. Steps 2 to 7 may be repeated, children collecting a different set from before.

activity 5 Match and mix: parts [Num 7.1/5] Agamefor2to5players.Itspurposeistoconsolidatefurthertheconceptsformed

in Activity 1.

Materials • Parts pack of cards.* • A card as illustrated below (provided in the photomasters). *The same as for Activity 4.

Rules of 1. The cards are spread out face downwards in the middle of the table. the game 2. TheMATCHandMIXcardisputwhereverconvenient. 3. Each player takes 5 cards (if 2 players only, they take 7 cards each). Alterna-

tively the cards may be dealt in the usual way. 4. They collect their cards in a pile face downwards. 5. Players in turn look at the top card in their piles, and put cards down next to

cardsalreadythere(afterthefirst)accordingtothefollowingthreerules. (i) Cards must match or be different, according as they are put next to each

other in the ‘match’ or ‘mix’ directions. (ii) Not more than 3 cards may be put together in either direction. (iii) There may not be two 3’s next to each other. (‘Match’ or ‘mix’ refers to the kind of part.) Examples (using A, B, C . . . for different kinds of part). A typical arrangement.

Num 7.1 Making equal parts (cont.)

c a a aB B B D DD B a

226

Num 7.1 Making equal parts (cont.)

aBcD

BBBB cccDDD

aaBBcc

aBa

Discussion of activities

None of these is allowed.

5. A card which cannot be played is replaced at the bottom of the pile. 6. Scoring is as follows. 1 point for completing a row or column of three. 2 points for putting down a card which simultaneously matches one way and

mixes the other way. 2pointsforbeingthefirst‘out.’ 1 point for being the second ‘out.’ (So it is possible to score up to 6 points in a single turn.) 7. Play continues until all players have put down all their cards. 8. Another round is then played.

Activities 1, 2 and 3 make use of Mode 1 schema building (physical experi-ence). This is every bit as important when introducing older children to fractions asphysicalsortingiswhenintroducingyoungchildrentotheconceptofaset.Inthe present case, it is the physical action of cutting up into equal parts from which wewantthechildrentoabstractthemathematicaloperationofdivision.Inthiscase, the operand is a whole object, so it is important to have something which can easilybecutup.Plasticineisidealforthis,sincechildrencanfirstmaketherightshapes and then cut them up. Activities 4 and 5 move on to pictorial representation of these physical actions. Thisiswheremosttextbooksbegin.Whatisnotunderstoodisthatthediagramscondense no less than six ideas, as will be seen in Num 7.3. These diagrams are likely to be helpful if and only if the children already have the right foundation schematowhichthesecanbeassimilated.Otherwise,hereisthefirstplacewherechildren can get confused. So in this topic, we begin with physical actions on objects: making equal parts. Wethenintroducediagramsrepresentingtheseandnomore.Wearenotyetintofractions; just objects, and parts of objects. The key ideas are that the parts must all be equal in size (or we call them ‘bits’); and that the kind of parts we are talking about depends only on how many the object is cut up into, not on their shape, nor onwhattheobjectis.Theseideasareencounteredfirstwithphysicalobjects,thenwith diagrams. OBserVe aND listeN reFleCt DisCuss

227

Num 7.2 take a NuMBer OF like parts

Concept That of a number of like parts.

Abilities (i) To put together any required number of any required kind of part. (ii) To recognize and name any such combination.

Thisisthenextsteptowardstheconceptofafraction.Itismuchmorestraightfor-ward than that of topic 1, which entailed (i) separating a single object into part-objects (ii) of a given number (iii) all of the same amount. Here we only have to put together a given number of these parts.

activity 1 Feeding the animals [Num 7.2/1]

Anactivityforupto6childrenworkingin2teams.Itspurposeistointroducetheconcept described above.

Materials • 3 sets of animals.* • Menu for each set, on separate cards.* • Cards for each menu showing standard sizes of eel, meat slab, biscuit. (One

between 2 children.) ** • 5 food trays. • Plasticine. • Blunt knives. • Cutting boards. • 5 set loops. * As listed below and provided in the photomasters. The animals may be models, or

pictures on cards. Quite a lot of these are needed; sorry, but if they help children to like fractions it’s worth it!

** A template for the meat slab is useful. See Num 7.1/1 under ‘Materials.’

SET 1 Animals 2 bears, 3 walruses, 5 seals, 6 penguins, 8 otters.

Menu

Thesealleatfish.

Today, the menu is eels (beheaded and tailed). Each bear gets the whole of an eel. Each walrus gets half of an eel. Each seal gets a third-part of an eel. Each penguin gets a fourth-part of an eel. Eachottergetsafifth-partofaneel.

Discussion of concept

228

SET 2 Animals 2 lions, 2 cheetahs, 4 wolves, 4 hyenas, 6 wild cats.

Menu

These all eat raw meat, supplied in large slabs.

Each lion gets the whole of a slab. Each cheetah gets half of a slab. Each wolf gets a third-part of a slab. Each hyena gets a fourth-part of a slab. Eachwildcatgetsafifth-partofaslab.

SET 3

Animals 3 rats, 3 voles, 5 shrews, 5 house mice, 7 harvest mice.

Menu

These all eat biscuits.

Each rat gets the whole of a biscuit. Each vole gets half of a biscuit. Each shrew gets a third-part of a biscuit. Each house mouse gets a fourth-part of a biscuit. Eachharvestmousegetsafifth-partofabiscuit.

What they do 1. One team acts as animal keepers, the other works in the zoo kitchen. The latter need to be more numerous, since there is more work for them to do.

2. A set of animals is chosen. Suppose that this is Set 1. The kitchen staff look at the menu and set to work, preparing eels, as in Num 7.1/1. The animal keepers put the animals in their enclosures (segregated, not assorted). They may choose how many of each.

3. The animal keepers, one at a time, come to the kitchen and ask for food for each animal in turn. The kitchen staff cut the eels as required. E.g.,

Animal keeper Zoo kitchen staff “Food for 3 walruses, please.” “Here it is: 3 halves of an eel.” “Foodfor4otters,please.” “4fifth-parts.Tellthemnottoleave any scraps.” “Food for 5 seals, please.” “Here it is: 5 third-parts of an eel.” “Food for 2 bears, please.” “Here you are : 2 whole eels. “Food for 3 penguins, please.” “3 fourth-parts. Lucky penguins.”

Num 7.2 Take a number of like parts (cont.)

229

Num 7.2 Take a number of like parts (cont.)

a standard eel

4. Each time, the animals’ keeper checks that the amounts are correct, and then gives its ration to each animal. The keepers also check each other.

5. Whenfeedingtimeisover,thefoodisreturnedtothekitchenforreprocessing.Steps 1 to 4 are then repeated with different animals, keepers, and kitchen staff.

6. Note that the eels, after their head and tails are removed, resemble the sausages of Num 7.1/1; and the slabs of meat are oblongs. Note also that the eels, slabs, and biscuits should be of standard sizes.

activity 2 Trainee keepers, qualified keepers [Num 7.2/2]

Anactivityforupto6children.Itspurposeistoconsolidatetheirrecognitionofthecombination: number of parts and kind of parts.

Materials • As for Activity 1, without the animals and set loops.

What they do Stage (a) Trainee keepers 1. Thechildrenareactingastraineekeepers.Inthispartoftheircourse,theyare

learning to ensure that they can recognize the right food for all their animals. 2. Initiallytheyworkwithonemenuatatime.Astandardcard(eel,meatslab,or

biscuit) is put at the top of the menu. 3. Each of them then makes up a food tray for a given number of animals of a par-

ticular kind. 4. Inturn,oneofthefoodtraysisputinthemiddleandtheotherssaywhatkindof

animal it is for, and how many. E.g.,

This is for 3 penguins.

5. Itisdifficulttojudgethedifferenceinsizebetween,e.g.,fourth-partsandfifth-parts. So the unused parts should be left on the cutting board, with any remain-ing Plasticine cleared away. Thus in the example above, the remaining fourth-part should be left on the cutting board.

6. As they become more expert, they work with 2, then 3 menus.

Stage (b) Qualified keepers 1. Whenoneofthemthinksheisreadytotakehistest,theothersprepare5trays

including at least 1 from each menu. 2. Iftheonetakinghistestcanidentifyallofthesecorrectly,hequalifies.(Apass

mark of 100% may seem severe, but every animal has to be correctly fed.)

230

activity 3 Head keepers [Num 7.2/3]

Anactivityforuptosixchildren.Itspurposeisforchildrentowritethetwoopera-tions which together go to make a fraction in their own words, as preparation for the standard notation which they will learn in the next topic.

Materials For each child: • The three menus already used, on one sheet of paper. • Lists of animals to be fed (see below). • Pencil and paper for each child.

What they do 1. A head keeper has to be able to write clear instructions for feeding any of the animals.

2. Those who wish to be considered for this job are each given one list of animals to be fed. Specimen lists are given below.

3. They write, in their own words, instructions for preparing a food tray for each cage of animals on their list.

4. Whentheyhavefinished,allbutoneofthechildrenactasexaminers.There-maining one reads out the questions, and his answers. The others decide wheth-er these are both clear and accurate.

5. Ifalltheanswersmeettheserequirements,thecandidateiseligibleforheadkeeper when there is a vacancy.

Writeclearandaccurateinstructionstellingsomeonehowtoprepareafoodtrayforeach of the following cages of animals:

LISTA LISTB LISTC

3 house mice 2 seals 1 vole 2 rats 2 otters 3 harvest mice 2 cheetahs 3 house mice 4 wolves 6 hyenas 4 voles 2 hyenas 1 seal 7 wild cats 3 bears 4 penguins 3 lions 5 walruses

LISTD LISTE LISTF

3 wildcats 3 bears 3 cheetahs 4 lions 1 penguin 2 hyenas 1 shrew 2 wolves 5 walruses 5 voles 8 wild cats 2 otters 1 bear 3 voles 5 rats 2 penguins 5 shrews 4 shrews

Num 7.2 Take a number of like parts (cont.)

231

Discussion of activities

Inthistopicwebringinthesecondcontributortotheconceptoffraction,return-ing for this to the same physical embodiments as in Num 7.1/1. As in the previous activity, these physical embodiments are of two kinds. The Plasticine is of the firstkind:aphysicaloperand,indifferentshapesrepresentingdifferentimaginaryoperands. The actions of making a given number of equal parts, and taking a given numberofthese,areofthesecondkind.Itisthesecondkindwhichleadstothemathematical concept of a fraction, independently of what the operand is. However,afractionhastohavesomeoperand.Soinstep3ofActivity1,Ihavekept in the words ‘. . . of an eel’ (etc.) to help us all to remember this. (You can skip the rest of this paragraph if you like.) This leads to a slight problem of syn-tax.Should‘eel’besingularorplural?Afterthought,Ihavesettledforsingular.‘Third-partofaneel’leadsto‘5third-partsofaneel.’Itistruethattheywon’tallcome from the same eel – but each part will come from an eel, not several. InActivity1,thefractionalpartsweretheend-pointoftheactivity,leadingtotheconcept.InActivity2,theyarethestartingpoint.Childrenhavetousetheirnewly formed concepts to recognize what number of what kind of parts they are lookingat.Theirdecisionsareconfirmedorotherwisebytherestofthegroup(Mode 2 testing). Activity 3 requires them to think about and describe the physical actions of making equal parts, and then taking a given number of these, instead of actually doing these actions. This takes them a step further towards fractions as mental operations.

OBserVe aND listeN reFleCt DisCuss

Num 7.2 Take a number of like parts (cont.)

Head keepers [Num 7.2/3]

232

Num 7.3 FraCtiONs as a DOuBle OperatiON; NOtatiON

Concepts (i) A fraction as a mathematical operation which corresponds to two actions: mak-ing a given number of equal parts, and putting together a given number of these.

(ii) A notation which represents this double operation.

Abilities (i) To match this operation (as represented by its notation) with a physical embodiment.

(ii) To match this operation (as represented by its notation) with a diagram for the same fraction.

As soon as we are dealing with mathematical operations, which are purely mental, we require a notation by which to communicate, manipulate, and record these. So the introduction of a notation, and the transition to the concept of a fraction as a combined operation, come together. The traditional notation is a good one,

how many parts of a given kind (numerator) what kind of parts (denominator)

Whatkindofpartsisdeterminedbyhowmanyequalpartsaremadefromawhole,soanumberissufficienttospecifywhatkindofpart.Thisisimpliedbythewaywereadit,“twothirds”;orasIthinkweshouldcontinuetosayatthisstage,“twothird-parts.” As has been said already, ‘part’ implies part ofsomething.Itimpliesanoper-and. There are now two further points to be made.

(i)Inthepresenttreatmentoffractions,thenumeratorisnottheoperand,and does not mean the same as 2 ÷ 3. (The latter meaning is also correct, but tointroduceitatthisstagecomplicatestheconceptfurther,andIamtryingtosimplify.)(ii) The whole is not the operand either. The operand is what this whole is whole of.

The traditional way of representing fractions by diagrams is also good, but very condensed.

This diagram for

simultaneously represents no less than six ideas:

12

23

Discussion of concepts

233

the object we start with (the operand);

the number of cuts we make, and where we make them (cuts as actions);

the cuts we have made (cuts as results), and the parts which result;

and the number of parts which we are taking.

So it is small wonder that children have problems when their learning of fractions begins with these diagrams. Intopic1,thesediagramswereintroducedafterchildrenhadmetwiththeseideas separately, in physical form. Here we shall renew the connection, and link it with the notation.

activity 1 Expanding the diagram [Num 7.3/1]

Anactivityforuptosixchildren,workingsinglyorintwosorthrees.Itspurposeisto relate the conventional fraction diagrams to the six concepts which they combine.

Materials • SARAR board, see Figure 20. • Pack of fraction diagram cards, see Figure 21.* • Reminder card (as illustrated below).* • Plasticine. • Knives. * Figure 21 shows two specimen cards. A full pack is provided in the Volume 2a

photomasters. You can invent more, if further practice is needed.

REMINDERCARDFOREXPANDINGTHEDIAGRAM

The diagram shows:- the object we start with first action the cuts we make first result the parts we make second action the parts we take combined result a fraction of an object

Num 7.3 Fractions as a double operation; notation (cont.)

234

What they do 1. The SARAR board is put where all can see it. To begin with, all except the dia-gram at the top is covered

2. Explain that this diagram shows the result of two actions combined. This combi-nation is called a fraction.

3. Uncover the rest of the board, and help them to work through the 5 steps shown. Help them to see how each step is embedded in the combined diagram. Leave thisonviewwiththefinalresultinPlasticinestillinposition.

4. Choose a suitable fraction diagram card. Help them to work through the steps which it represents, using the reminder card as a guide. The Plasticine objects are put in the blank half of the card. Repeat this step if necessary.

5. Whentheyarereadytocontinue,eachpairortriotakesafractioncardfromthe set. They may choose cards they think they can do. Note that the long thin oblong may be modelled as a sausage, and the torus as a ring doughnut. The children may use either description.

6. For each card, they work through the steps which this represents, agreeing each step with each other.

7. Whenfinished,theyagreewhattheresultiscalled.(E.g.,2fifth-partsofaringdoughnut.)

8. Steps5,6,7arerepeateduntiltheyareconfidentthattheycaninterpreteverypart of the meaning of a fraction diagram.

start first action first resultcombined

resultsecond actionTake 2 of these

Num 7.3 Fractions as a double operation; notation (cont.)

A Plasticineslab

Cut it alongthese lines

3third-partsof a slab

Discardwhat is left

2third-partsof a slab

Figure 20 SARAR board for ‘Expanding the diagram’

Figure 21 Specimen fraction diagram cards

235

activity 2 “Please may i have?” (Diagrams and notation) [Num 7.3/2]

Agamefor5or6players.Itspurposeistolinkthediagramswiththeconventionalnotation.

Materials • Set of fraction diagram cards.* • Set of fraction notation cards.** * As used for Activity 1. **Illustratedinstep2belowandprovidedinthephotomasters.

1. Allthecardsaredealttotheplayers,eachpackseparately.Ifthereare5players,eachwillthusget3cardsofeachsort.Ifthereare6players,eachwillreceive2cards of one sort and 3 of the other.

2. The purpose is to get rid of one’s cards by making pairs. Each pair must consist of a notation card and a diagram card. E.g.,

3. Afterthedeal,theplayersfirstputdownanypairswhichtheyalreadyhold. 4. They then try to make more pairs by asking each other, taking turns. E.g.,

“Please,Susan,mayIhaveadiagramfortwothird-parts?”Or,“...anotationfor two third-parts?”

5. The winner is the player who has most pairs when all have played out their hands.

Rules of the game

35

35

Read as

3

fifth-parts

Num 7.3 Fractions as a double operation; notation (cont.)

236

Becausesomanychildren(andadults)havedifficultywithfractions,Ithinkthatit is important to continue with physical embodiments of both the actions and the operands in order to give plenty of Mode 1 learning to establish a good founda-tion. This is particularly important when, as in this case, diagrams are used which themselves are highly condensed. So in Activity 1 we return to physical actions on physical operands. The notation, which represents the same concept as a fraction diagram, must therefore inevitably be more condensed even than the diagram. So the children need–andIhopebynowwillhavehad–plentyofexperienceoftheseparateoperations, before they learn the conventional notation. InActivity2,thewords‘of...’havebeendroppedfortworeasons.Thesameoperator can act on a variety of operands, so we have now begun to treat fractions independently of any particular embodiment.

OBserVe aND listeN reFleCt DisCuss

Discussion of activities

Num 7.3 Fractions as a double operation; notation (cont.)

“Please may I have?” (Diagrams and notation) [Num 7.3/2]

237

Num 7.4 siMple equiValeNt FraCtiONs

Concept That different fractions of the same operand may give the same amount.

Ability To recognize when this is the case, i.e., to match equivalent fractions.

2 fourth-partsof a biscuit

and

1 halfof a biscuit

are not identical, but we get the same amount to eat (assuming no crumbs). So we say that and are equivalent fractions, meaning that if applied to the same oper-and they result in the same amount: though these may not look exactly alike. The idea of equivalence is an important one throughout mathematics. (For a fuller discussion, see chapter 10 of The Psychology of Learning Mathematics).Itis important in this network because it makes the bridge between fractions as dou-ble operations, and fractions as representing numbers. Itistheideawhichisimportant,andinthistopicIhavethereforekepttosimpleexamples.

activity 1 “Will this do instead?” [Num 7.4/1]

Anactivityforupto6children.Itspurposeistointroducetheconceptofequivalentfractions.Italsointroduceseighth-parts.

Materials • Plasticine. • 8 card templates for fruit bars (See SAIL Volume 2a, Photomaster 156). • Knife. • Board to cut on. • A small rolling pin is also useful.

What they do 1. One of the children acts as the owner of a candy store. He sells very good fruit bars.

2. Before the store opens, he makes a number of these, each the size of one of the cards. The cards are used as supports.

(Initiallyyoumayfinditbettertodothisyourself.Ithelpstopreventstickingifthe cutting board and rolling pin are wetted).

Discussion of concept

24

12

238

3. Fruit bars are expensive, and pennies are scarce. So the store owner, a kind old man, cuts them up so that children can buy them in parts. Two he cuts into halves, two into fourth-parts, and two into eighth-parts. (This leaves two spare cards.)

4. He is also eccentric, and tries never to give exactly what is asked for. 5. Whenthestoreopens,acustomercomesinandasksfor(e.g.)halfofafruitbar.

The store owner takes 2 fourth-parts, or 4 eighth-parts, and puts these on a spare card.Heoffersthesetothecustomer,saying“Willthesedoinstead?”

6. The customer says (in this case) “2 fourth parts – that will do nicely, thank you.” (Or “4 eighth-parts . . . ,” as the case may be.)

7. Another customer enters the store, and steps 5 and 6 are repeated. 8. The stock is replenished when necessary. 9. Sometimesthestoreownermakesamistake.(Ifthisdoesnotarisenaturally,he

doesso‘accidentallyonpurpose.’)Inthiscasethecustomerpolitelyexplains. 10. The game may be extended by the inclusion of other goodies, of various shapes.

activity 2 sorting equivalent fractions [Num 7.4/2]

Anactivityforupto6children.Itspurposeistoexpandtheconceptofequivalentfractions to diagrams and notation.

Materials • Equivalent fractions pack 1 (see specimens below and the photomasters). • 3 set loops

What they do 1. Thefractionpackisshuffledandspreadoutfaceupwardsonthetable. 2. Children work together in ones or twos, according to how many there are. Each

child (or pair) takes a set loop. 3. To begin with, the notation card for a whole, 1 half, and 1 fourth-part are found.

One is put in each set loop. 4. They then collect all fractions which are equivalent to this, whether diagrams or

notation. 5. They check each others’ sets, and discuss if necessary. 6. As a continuation activity, each equivalence set may be sorted into subsets. E.g.,

within the set of fractions equivalent to one half, one subset would be all the diagrams representing the fraction 2 fourth-parts, together with its notation.

Num 7.4 Simple equivalent fractions (cont.)

24

12

12

1 21248

14

24

2 424

48

4 848

14

1 414

239

activity 3 Match and mix: equivalent fractions [Num 7.4/3]

Anactivityfor2to5players.Itspurposeistogivefurtherpracticeinrecognizingequivalent fractions.

Materials • Equivalent fractions pack 1.* • Match and mix card.** * The same as for Activity 2. ** Like that for Num 7.1/5.

This is played in the same way as Match and mix: parts (Num 7.1/5). Here, ‘match,’ or ‘mix’ refers to equivalence of fractions, as represented by diagrams and notation.

InActivity1,thephysicalembodimentsprovideagoodMode1refresherofearlierwork, and foundation for the concept of equivalent fractions. And the imagined situation embodies one of the meanings of equivalent (equi-valent: equal in value). Equivalent fractions of a fruit bar contain the same amount to eat, and one worth the same amount of money. Youwillnoticetheomissionofthird-parts.Whilethesefitwellintotheschemaassofardeveloped,Iseenoobjectinchildren’slearningaboutsixth-partsjusttoprovide something two of which are equivalent to third-parts. And looking for-wardtotheequivalentdecimalfractions,Ihavelittleenthusiasmforteachingchil-dren recurring decimals just in order to keep going with third-parts. Eighth-parts, on the other hand, form part of the sequence formed by successive halving and arestillwithusineverydaylife.BoththerulersonmydeskasIwritestillhaveinches along one edge, and these are subdivided into halves, quarters, eighth-parts, and sixteenth-parts. InActivity2,theyextendthenewconcepttothemoreabstractrepresentationsof fractions provided by diagrams and notation. They have already learned the connections between these, in Num 7.3/2. Activity 3 consolidates these in a game which they have already learned to play, with simpler materials. Iwouldnotencourageteachingthecancellingrule,whichisaconvenienttech-nique but does not relate clearly to the concepts. Children will themselves notice the inverse, that equivalent fractions may be obtained by doubling numerators and denominator. This corresponds to cutting every part in two.

OBserVe aND listeN reFleCt DisCuss

Discussion of activities

Rules of the game

Num 7.4 Simple equivalent fractions (cont.)

240

Num 7.5 DeCiMal FraCtiONs aND equiValeNts

Concepts (i) Decimalfractions. (ii) Equivalence between decimal fractions and fractions of other denominators.

Abilities (i) Tofinddecimalfractionsofavarietyofoperands. (ii) To match decimal fractions with other equivalent fractions.

The common usage of the term ‘decimals’ confuses (i) a particular kind of frac-tions, and (ii) place-value notation. Unless we separate these two ideas, we would havetosaythat0.5isadecimal,while5/10isnot.SoIhopethatyouwillacceptmy proposal, that we use the term ‘decimal’ to mean what it says, i.e., related to 10 (as in ‘decimal coinage’); and ‘decimal fraction’ to mean a fraction whose denomi-natorisapowerof10(10,100,1000...).whicheverwayitiswritten.Ifthisisagreed, then the term ‘common fraction’ (or ‘vulgar fraction’) is no longer a useful one. Decimalfractionshavethegreatadvantagethattheycanberepresentedintwonotations: bar notation (the one used in this network up to now), and place-value notation.Thelatteroffersconsiderablesimplificationwhenitcomestocalcula-tions – not least, that we can use a calculator! Inthistopic,weshallconfineourselvestotheintroductionofdecimalfractions,and their equivalences with fractions of other denominators. Their representation in place-value notation is a separate topic in itself.

activity 1 Making jewellery to order [Num 7.5/1]

Anactivityforupto6children.Itspurposeistointroducedecimalfractionsinver-bal form.

Materials • A metre measure, divided in decimetres and centimetres. • Jewellery catalogue, see Figure 22.* • Purchasing guide, see Figure 23.* * Also in photomasters

What they do 1. (Introductorydiscussion.)Theyalreadyknowthat100cmisthesamelengthas1m;sotheyshouldfinditeasytoanswerthequestion“Whatisahundredth-part of a metre?” Next, they learn that the other length on the metre measure is called a decimetre, because there are ten of these in a metre. So a tenth-part of a metre is called . . .?

2. Two children take the roles of jeweller and jewellers’ supply merchant. The oth-ers in turn act as customers.

3. The jeweller works at home, making jewellery to order. She has a catalogue to show to the customers, and a purchasing guide for her own use.

4. Thefirstcustomercalls,looksatthecatalogue,andchooseswhatshewouldlike.E.g., a coiled brooch.

Discussion of concepts

241

Num 7.5 Decimal fractions and equivalents (cont.)

Figure 22 Jewellery catalogue

Figure 23 Purchasing guide

242

Suggested sequence for the discussion

Num 7.5 Decimal fractions and equivalents (cont.)

5. The jeweller looks at her purchasing guide, and sees that a metre of silver wire willmakefiveofthese.Sheonlywantsenoughforone.

6. Soshegoestothesupplierandasksforonefifth-partofametreofsilverwire. 7. Thesupplierlooksathermetremeasure,andsays“I’llmeasureyou2decime-

tres; that is, 2 tenth-parts of a metre.” 8. Thejewellersays“Iagree,”orifshehasanydoubts,thematterisdiscussedwith

the help of the metre measure. 9. The jeweller returns home, another customer calls, and steps 4 to 8 are repeated. 10. Whenalltheremainingchildrenhaveactedascustomers,twomorechildren

take the roles of jeweller and supplier, and steps 4 to 8 are repeated. 11. Notethatsinceafiligreependanttakesafourth-partofametre,thesupplierin

thiscasesays“I’llmeasureyou25centimetres;thatis,25hundredth-partsofa metre.” For variety, centimetres may be used on other occasions. E.g., for a coiledbrooch,thesuppliercouldalsosay“I’llmeasureyou20centimetres;thatis, 20 hundredth parts of a metre.”

activity 2 Equivalent fraction diagrams (decimal) [Num 7.5/2]

Ateacher-leddiscussionforupto6children.Itspurposesare(i)tomakeexplicitthe concepts of a decimal fraction, introduced in Activity 1, (ii) to extend their use of fraction diagrams to those for decimal fractions, (iii) to use these diagrams for rec-ognizing equivalences between decimal and non-decimal fractions, and also between decimal fractions with different denominators.

Materials • For each child, a pair of equivalent fraction diagrams (decimal), as illustrated below and provided in the photomasters. These are best on separate cards.

• Something to point with.

1. Giveeachchildapairofequivalencediagrams.Itdoesn’tmatterwhichwayuptheylook at these. Begin by asking them to look at the one shown above on the left.

2. Whatkindofpartsarerepresentedbythenarrowestoblongs?(Tenth-parts.)By the oblongs between the thicker lines? (Fifth-parts.) And by the oblongs marked off by the arrowed line? (Halves.)

3. Whatequivalencescanwededuce?(1fifth-partisequivalentto2tenth-parts.2fifth-partsareequivalentto4tenth-parts,etc.And5tenth-partsareequivalenttoone half.)

v

v

243

Num 7.5 Decimal fractions and equivalents (cont.)

4. Next,askthemtolookattheotherdiagram.Discussthisinasimilarway.(Wecan deduce that 1 fourth-part is equivalent to 25 hundredth-parts, etc. And that 1 half is equivalent to 50 hundredth-parts.)

5. Next,tellthemtoputthetwodiagramssidebyside.Whatcantheysaybycomparingthese?(1tenth-partisequivalentto10hundredth-parts.1fifth-partis equivalent to 20 hundredth-parts. Also, 3 tenth-parts are equivalent to 30 hundredth-parts,etc.And2fifth-partsareequivalentto40hundredth-parts,etc.)

6. Finally, tell them that there is a special name for the kind of fractions they have been talking about: decimal fractions. These include tenth-parts and hundredth-parts.Cantheythinkofanotherdecimalfraction?(Whatpartofametreisamillimetre?Whatpartofakilometreisametre?)

activity 3 Pair, and explain [Num 7.5/3]

Agameforupto6children.Itspurposeistorelatetheequivalencetheyhavelearned in Activity 2 to the notation which is common to all kinds of fraction (com-mon fraction notation).

Materials • Fractions, pack 2 (see specimens below).* • For each child, a pair of equivalent fraction diagrams (decimal).** * A full set is provided in the photomasters ** The same as used in Activity 2.

1. Thefractionspackisshuffledandputfacedownonthetable.Thetopcardis

turned face up and up and put separately. 2. Inturn,eachplayertakesthetopcardfromthepileandputsitfaceuponthe

table. The face-up cards are spread out so that all can be seen at the same time. 3. The object is to form pairs of equivalent fractions. These will gradually show as

more cards are turned face-up. Pairs may be made from identical fractions since these too are equivalent.

4. Theplayerwhohasjustturnedacardhasfirstchancetomakeapair.Shetakesthese, puts them side by side in front of her, and explains the equivalence in termsoftheequivalencediagrams.Iftheothersagree,shekeepsthepair.Ifnot, there is discussion.

5. Ifapairisoverlooked,theplayerwhosenextturnitismayclaimitbeforeturn-ingoveracard.Shethenturnsoveracardasinstep4.Ifanotherpairresults,she may take this also.

6. Whentherearenomorecardstoturnoverthewinneristheplayerwhohasmostpairs.

Note Theexplanationinstep4isaveryimportantpartofthisactivity.Itmustbeintermsof the diagrams; not in terms of the cancelling rule, which is a convenient shortcut but makes no contribution to understanding.

24

12

12

1 212

24

2 424

50 100

50 100

25 100

25 100

50

100

50 100

25 100

25

100

Rules of the game

244

Rules of the game

activity 4 Match and mix: equivalent decimal fractions [Num 7.5/4]

Agamefor2to5players.Itspurposeistogivefurtherpracticeinrecognizingdeci-mal equivalent fractions.

Materials • Fractions pack 2.* • Match and mix card.** * The same as used for Activity 2.* ** The same as used for Num 7.1/5 and Num 7.4/3.

This is played in the same way as ‘Match and mix: parts’ (Num 7.1/5), but ‘Fractions pack 2’ is used and ‘match’ or ‘mix’ refers to the equivalence of decimal fractions in bar notation.

Inthistopicwearestillusingbarnotation,forbothdecimalandnon-decimalfrac-tions.(See‘Discussionofconcepts.’) InActivity1,themetre,subdividedintotenth-parts,hundredth-parts,andthou-sandth-parts, provides a familiar and useful physical embodiment of the new con-ceptstobelearned.Ihaveintroducedthedecimetreherefortworeasons.First,because we need it as a representation of the fraction a tenth-part; and second, because it is a useful unit which in my view deserves to be more widely used than it is. InActivity2,thenewconceptismademoreexplicitbydiscussion(Mode2),and related to equivalence diagrams like those already used for non-decimal frac-tions.NonewmaterialisintroducedinActivity3.Itconsolidatesthenewcon-cepts by Mode 2 testing. As we all know, there is nothing so good as trying to explain something to others, to make one get it clear in one’s own mind. Activities 3 and 4 work entirely with symbols, and so complete the process by which a con-cept is detached from any particular physical embodiment.

OBserVe aND listeN reFleCt DisCuss

Num 7.5 Decimal fractions and equivalents (cont.)

Discussion of activities

Match and mix: equivalent decimal fractions [Num 7.5/4]

245

Num 7.6 DeCiMal FraCtiONs iN plaCe-Value NOtatiON

Concept Extension of place-value notation to represent decimal fractions.

Abilities (i) To represent, and recognize, decimal fractions in place-value notation. (ii) To recognize the same fraction written in place-value and in common fraction

notation.

Weareverydependentonnotationsofmanykindsnotonlyforcommunicatingand recording our ideas, but as a help to the thinking process itself. Some nota-tions do their job much better than others. Place-value notation is a very good one, so the combination of decimal fractions and place-value notation is a very useful one.Withtheadventofcalculatorsandcomputers,Iseenoreasonforchildrentolearn to calculate in bar notation until such time as they are about to learn algebraic fractions.

Note that we are here talking about equal fractions.These

and .04

are two notations for the same fraction.

Note Children should already have completed topics NuSp 1.9 (interpolation between points on a number line) and NuSp 1.10 (extrapolation of place-value notation) be-fore beginning the activities which follow.

activity 1 reading headed columns in two ways [Num 7.6/1]

An activity for up to 3 children. (They all need to see the activity board the same wayup.)Itspurposeistorefreshtheirknowledgeofplace-valuenotationfordeci-mal fractions, and give additional practice in pairing the two notations for decimal fractions.

Materials • Activity board.* • Number cards.** * See illustration in step 4 following. The full version is in the Volume 2a photomas-

ters. Two should be provided, so that up to 6 children can play at a time. ** 1 to 9, 3 cm square, 2 of each (see photomasters).

4100

Discussion of concept

246

Num 7.6 Decimal fractions in place-value notation (cont.)

What they do 1. To introduce the activity, show the children the familiar headed columns.

H T Ones

Starting at the ones, each column we go to the left means that the numbers are 10 times bigger. So what would it mean if we had some more columns on the right of the ones column? Like this.

H T Ones

Whataretentimessmallerthanones?Tenth-parts.Andwhataretentimessmaller than these? Hundredth-parts. To emphasize the boundary between wholes and parts, we will use a dotted line. So we now have headed columns like these, using t-p and h-p for tenth-parts and hundredth-parts. (Note the use of lower case letters for parts of a unit, as in the metric system of abbreviations.)

H T Ones t-p h-p

Wecanhaveasmanycolumnsaswelike,extendingtotheleftandrightofthedotted line. For the present activity, three columns are enough. (See step 4 il-lustration.)

2. The activity board is then put on the table where all the children can see it right wayup.Thenumbercardsareshuffledandputinapile(orheap)facedown.

3. Inturn,thechildrentakenumbercardsfromthepileandputthemfaceupwhereindicated by squares on the activity board. They take one, two, or three cards as neededtofillaline.

4. Theythenreadtheresultingfractionintwoways,asintheexamplesbelow.(Itdoesnotmatterwhichissaidfirst,anditisagoodideatovarythis.)

Ones t-p h-p

7

3

4 5

First child: “7 tenth-parts. Zero point 7 zero.” Second child: “3 hundredth-parts. Zero point zero 3.” Third child: “4 tenth-parts and 5 hundredth-parts. Zero point 4,5.” (“Four,five”not“forty-five.”) (Next time around, they could speak these the other way about.) 5. At this stage a zero is spoken for every empty column. This is never incorrect.

Sometimes it is essential as a place-holder, sometimes it is not needed. This is a separate question which is dealt with in the next activity.

6. Whenallthelinesarefilled,theboardisclearedandsteps3and4repeated.

247

activity 2 same number, or different? [Num 7.6/2]

Anactivityforupto3children.Itspurposeisforthemtolearnwhenazeroises-sential as a place-holder, and when it is optional.

Materials • Activity board.* • Number cards.* • Pencil and paper for each child. * The same as for Activity 1.

What they do 1. The activity board and number cards are put out as for Activity 1. 2. ThefirstchildputsnumbercardsontheboardasinActivity1. 3. He then reads these in two ways, with and without zeros for the empty columns. 4. The other two write exactly what is said, both ways. They then decide whether

each of these is the same number as the one on the board, or different. 5. Steps 2, 3, 4 are repeated, the next child putting down the cards and reading

them. 6. Examples:

(Top line) Spoken: “Zero point 6 zero. Point 6.” Written: 0.60 .6 same number same number

(Second line) Spoken: “Zero point zero 2. Point 2.” Written: 0.02 .2 same number different number

7. Thiscontinuesuntiltheboardisfilled. 8. Steps 2 to 7 may be repeated. 9. They may then try to put into words what determines the outcome – i.e., when

zeros are necessary for place-value notation to represent correctly the fraction given by the headed columns, and when they are optional. This is not very easy, soIsuggestitassuitableonlyforthebrighterchildren.

10. A concise formulation is: “Leading and trailing zeros are optional. Sandwiched zeros are necessary as place-holders.” (‘Sandwiched’ means between two digits, or a digit and the decimal point.) This is a convenient rule, but children should beabletogivereasonsaswell.Norshouldtheybegiventhistomemorize.Ifthey can produce a correct formulation of their own all the better.

Ones t-p h-p

6

2

Num 7.6 Decimal fractions in place-value notation (cont.)

248

Notes (i)Inthecontextofmeasurement,.6mand.60mdohavedifferentmeanings.Thefirstmeansthatwearemeasuringtothenearestdecimetre;thesecond,tothenearestcentimetre, the result being .60 as against .61 or .62.

(ii) Many people prefer to write and speak (e.g.) 0.4 rather than .4, on the grounds thatthishelpstopreventthedecimalpointfrombeingoverlooked.Iseethisasapurely personal preference. Mathematically, both stand for the same number, and this is what the present activity is about.

activity 3 claiming and naming [Num 7.6/3]

Agamefor5or6players.Itspurposeistogivepracticeinrecognizingthesamedecimal fraction written in two notations, bar and place-value.

Materials • Fractions,packsPVandB.* * See specimens shown in Figure 21. (A complete set of each is provided in the pho-

tomasters.).PackPVisinplace-valuenotation,packBinbarnotation.Alimitedrange of digits is used to reduce the cues from these.

Figure 21 Fractions, packs PV and B (specimens)

1. PackPVisshuffledanddealttotheplayers.PackBisshuffledandputfacedown on the table.

2. Players look at their hands. 3. Each player in turn picks up the top card from the pack on the table and puts it

face up. 4. Whicheverplayerhasinhishandthecardforthesamefractionsays“Iclaim

that”, and takes it. 5. He puts the two cards together in front of him, and names both aloud. E.g., “4

hundredth-parts. Point zero 4.” The others check. 6. Thegamecontinuesasinsteps3,4,5.Thefirsttoputdownallhiscardsisthe

winner, but the others play out their hands. 7. Next time the game is played, the packs are used the other way about. That is, B

isdealttotheplayersandPVisputonthetable.

Num 7.6 Decimal fractions in place-value notation (cont.)

.2 210

.2 210

PV B

Rules of the game

249

Wearenowworkingatapurelyabstractstage,withoutthesupportofphysicalmaterials or diagrams. So the earlier work, devoted to establishing the concepts firmlywiththehelpoftheabove,isofgreatimportanceinpreparationforthis.Ifchildrenhavedifficultyatthisorlaterstages,thentheyneedmoretimeworkingatthe more concrete levels. This topic is concerned with notations rather than concepts (which emphasizes further the need for the concepts to have already been well established). Headed column notation continues to provide a way of showing what the numbers stand for which is clearer and more explicit than place-value notation, but which can be translated easily to and from place value notation. Bar notation provides a link with the earlier work on fractions of all denominators. And the expansion of place-value notation to represent fractions implies also the expansion of the concept of number to include fractions. This implication will be made explicit in the next topic.

OBserVe aND listeN reFleCt DisCuss

Num 7.6 Decimal fractions in place-value notation (cont.)

Discussion of activities

Claiming and naming [Num 7.6/3]

250

Num 7.7 FraCtiONs as NuMBers. aDDitiON OF DeCiMal FraCtiONs iN plaCe-Value NOtatiON

Concept Fractions as a new kind of number.

Ability To add decimal fractions, using place-value notation.

Fractions are often treated as numbers right from the start. This almost guarantees confusion for the children, since they are seldom told that we are expanding our concept of number to include a new kind of number. Place-value notation makes this transition smoother, but does not remove the need to explain what is going on, so that children have a chance to adjust their thinking. Moreover, there are two distinct concepts involved, each with much interiority. Thefirstisthatoffractions–anewcategoryofmathematicalideas.Thesecondis that of other possible number systems. (Until now, children have only met one number system.) So one of the concerns of this topic is to introduce children to thinking of fractions as numbers. This we do in Activity 1. Here we show that fractions, in place-value notation, may be added in the same way as whole num-bers; and that the results make sense. So in this respect they ‘behave’ like the numbers with which we are already familiar. Whetherwearejustifiedinaddingfractionsinthesamewayasthefamiliarcountingnumbersisanotherquestion,andamoredifficultone. The problem is not unlike that of a naturalist who discovers a new species. First, he has to become aware of its existence. Then he has to decide how to cat-egorize it, in the same way as one has to decide whether a bat is a bird or a rodent. And the most important characteristic for classifying may not be the most obvious, whichinthiscaseisflight. To qualify for acceptance as a number system requires that these new (mental) objects ‘behave’ like the numbers we are already familiar with. The characteristics required by mathematicians are, again, not the most obvious. They have already been described in Num 5.10; namely, that addition is commutative and associative, likewise for multiplication, and multiplication is distributive over addition. Allthisis,ofcourse,farbeyondourpresentneeds,andImentionitforyourinformation rather than for your detailed consideration. Also, so that you will un-derstandwhyIdonottreatitasaseparatetopicforthechildren.Forthebrighterchildren,however,Ithinkitisworthtouchinglightlyonthisquestion,inthecon-text of addition only. This is done in Activity 2.

Discussion of concept

251

activity 1 target, 1 [Num 7.7/1]

Agameforupto3players(sincetheyneedtoseethecardsthesamewayup).Itspur-pose is to introduce the idea of adding decimal fractions, using place-value notation.

Materials • Tenth-part cards.* • Hundredth-part cards.* • Pencil and paper for each player. * See illustration in step 2 below and the photomasters.

Introduction 1. “Now that we are writing decimal fractions like this they look rather like the ordinary numbers we’re used to. Suppose we decide to treat them like numbers, how would we add them?” E.g.,

2. Follow this with other examples, including those which involve ‘carrying’; firstonlytotherightof the decimal point, e.g.,

3. Then across the decimal point, e.g.,

Easier version 1. Thetwopacksofcardsareshuffledandputseparately,facedown. 2. Each player in turn takes the top card from each pack. She may use one of these,

or both combined, the smaller card on top; or she may decide to use neither if she wishes.

3. Each player writes this number, and adds it to the previous number or total to give a cumulative total.

4. The used cards are replaced at the bottom of the pile. 5. The player who reaches 1 unit exactly is the winner. 6. Overshootingisnotallowed.Ifthenumbersturnedwouldtakethetotalpast1

however they are used, then they are not used. Harder version Thisisplayedasabove,exceptforstep4.Inthisversion,allthecardswhichhave

been turned over remain face up, and are spread out so that all are visible. The player whose turn it is may use any one, or two, of these. So at each turn, there are two more cards to choose from.

Num 7.7 Fractions as numbers. Addition of decimal fractions in place-value notation (cont.)

.47

.47+ .32

.74+ .52

.26+ .39

.0 8.3 .0 8 .3

Rules of the game

oror

252

activity 2 “How do we know that our method is still correct?” [Num 7.7/2]

Ateacher-leddiscussion,forthemoreablepupilsonly.Itspurposeisfirsttoputtheabove question, and then to answer it so far as addition is concerned.

Materials • 2 decimal unit squares, see Figure 22 and the photomasters. • Base 10 material. • Pencil and paper for each child.

1. Beginbyputtingthequestion,alongthefollowinglines.“Whenwestartedwrit-ing decimal fractions in place-value notation instead of bar notation, they looked much the same as ordinary whole numbers except for the decimal point. But of course they are not the same.”

2. Emphasize the difference by asking them to read each of the following in two ways.

“Thirty-seven. Three tens, seven ones.”

(Now insert a decimal point.)

“Point three seven. Three tenth-parts, seven hundredth-parts.”

Note Never tolerate “point thirty seven”, which guarantees self-confusion.

3. “So when we’ve been adding decimal fractions like ordinary whole numbers, we have assumed that the way of adding which is correct for ordinary whole numbers is still correct for fractions. How do we know that our method is still correct,nowthatthefiguresmeansomethingdifferent?”

Num 7.7 Fractions as numbers. Addition of decimal fractions in place-value notation (cont.)

Figure 25 Decimal unit square

37

.37

Suggested sequence for the discussion

253

4. Showadecimalunitsquare.“Ifeachofthelittlesquaresstandsforaunit,whatdoes the big square stand for?” (A hundred.) But if we change the meaning so that the big square stands for a unit, what does each little square now stand for?” (A hundredth-part.)

5. Put a rod from the base 10 materials in a column of the unit square. “And what does this now stand for?” (A tenth-part.)

6. So how can we represent (e.g.) .34 on this unit square? (3 rods, 4 cubes.) 7. The continuation to addition is now straightforward. The two numbers to be

addedarefirstrepresentedseparatelyonthetwounitsquares.Therodsandcubes are then put together on one square, and the result interpreted. Begin with examples which do not require carrying; then, carrying from hundredth-parts to tenth-parts; then, from tenth-parts to ones (whole numbers). You can continue, according to your own judgement, by combining both kinds of carrying.

8. Again, use your own judgement about how many examples the children need to use in order to establish the deeper meaning of children’s use of the symbols for adding fractions.

Inthistopic,weuseourextrapolationofplace-valuenotationtoextrapolatetheidea of number, from the familiar whole numbers to a new kind called fractional numbers. The word ‘fraction’ has now acquired a second meaning. This meaning is assumed in topic 1, where the familiar methods for adding are applied to numbers of this new kind. This is concept building (Mode 3). Provided thatthismakessensetothemintuitively,Ithinkthatformanychildrenitmaybebest to leave it at that. However, there will be some who can see the point of the question put in topic 2. For these, the same base 10 materials which helped to provide good conceptual foundations for the early number work in place-value notation are useful in helping them to strengthen their conceptual foundations for the new number work. This topic makes a start with Mode 3 testing, that these new concepts are consistent with our mathematical knowledge.

OBserVe aND listeN reFleCt DisCuss

Discussion of activities

Num 7.7 Fractions as numbers. Addition of decimal fractions in place-value notation (cont.)

254

Num 7.8 FraCtiONs as quOtieNts

Concepts (i) A fraction as a result of sharing. (ii) A fraction as a quotient.

Abilities (i) To share equally a number of objects greater than 1 when the result includes fractions of objects.

(ii) To predict the results of (i) as it would be shown in place-value notation by a calculator.

(iii) To use calculators to save time and labour, but thoughtfully.

Oneofthewaysinwhichmanytextbookscreateunnecessarydifficultiesforchil-dren is to confuse fractions as quotients with the other aspects of fractions which have already been discussed. This is like confusing the comparison and take-away aspects of subtraction; but worse, because fractions are harder, even when correctly explained. Ifwereadas‘twofifths,’shortfor‘twofifth-parts,’thecorrespondingphysi-calactionsare:startwithanobject,makefiveequalparts,taketwooftheseparts,resulttwofifth-parts. Now, however, we are going to think of in a different way, as the result of division. The corresponding physical actions are: take two objects, share equally amongfive,eachshareishowmuch? At a physical level these are quite different, just as physically 3 sets of 5 objects and 5 sets of 3 objects are quite different, and mathematically 3(5) and 5(3) are dif-ferent; although the results are the same. So we should be more surprised than we are that the division 2 ÷ 5 gives as result. And to teach that is just another way of writing 2 ÷ 5 is to beg the whole question. 2 ÷ 5 is a mathematical operation, a division. is the result of this op-eration, a quotient. The distinction becomes even sharper if we write this quotient as 0.4 . To put this another way: would we say that 45 was just another way of writing 5 x 9? Finally,Isuggestthatweshouldbemoresurprisedthanmostpeopleseemtobethataquotientisanumberatall.Itis,infact,anewkindofnumber:afractionalnumber. A full discussion of this expansion of our schema for numbers, and what makes it legitimate and useful, is beyond the scope of what we are doing here; nor isitnecessary.ButIhopethatthesequenceinwhichtheideashavebeenoffered,and the activities, will at least help to build a clearer understanding of the various aspects of fractions than children have usually been able to acquire.

25

25

25 2

5

25

Discussion of concepts

255

activity 1 Fractions for sharing [Num 7.8/1]

Anactivityforasmallgroupofchildren.Itspurposeistointroducethemtothecon-nection between fractions and the sharing aspect of division.

Materials • Plasticine. • Knives. Preferably • Cutting boards. one per two • Fruit bar templates.† children. • SAR board: fractions for sharing, see Figure 23.* • Number cards 2 to 5.** • Non-permanent marker. • Wiper. *Thefull-sizedonefromthephotomastersshouldbecoveredwithfilm. **Tofitthespacesontheboard,asprovidedinthephotomasters. † See SAIL Volume 2a, Photomaster 156.

What they do 1. About 5 Plasticine ‘fruit bars’ are made, using the cards as templates to give a uniform size.

2. To begin with, put a ‘2’ in the left-hand space and a ‘3’ in the right-hand space of theSARboard.Askthechildrentodoasitsays.Thefirsttimetheymayneedalittle help.

3. One way to do this is to put two fruit bars, one on top of the other, and cut these into3equalparts.Whentakenapart,eachshareis2third-partsofafruitbar.Thismayintroducepracticaldifficulties,e.g.,thelayersofPlasticinemaysticktogether if too warm; but conceptually it is probably the clearest.

4. So they should write a number sentence something like this:

“2 fruit bars, shared between 3, result of a fruit bar each.”

5. LetthemexperimentwithothernumbersintheSTARTandACTIONspaces.Tobegin with, the smaller number should be in the START space.

6. Whentheyfullyunderstandtheforegoinginphysicalterms,continuetostep7.This corresponds to the second instruction on the SAR board.

7. Remind them that sharing is one kind of division. So they can shorten their number sentences to (e.g.)

8. Thisisreadas“2dividedby5equals(or‘isequalto’)2fifth-parts.”Emphasizethat we are not saying that is another way of writing 2 ÷ 5.

The left-hand side says what is done.Itstandsformakingequalshares.Theright-hand side says how mucheachpersongets.Itiscalledaquotient.(Theword means ‘how much,’ or ‘how many.’)

Num 7.8 Fractions as quotients (cont.)

}

23

25

2 ÷ 5 = 2 5

256

Figure 26 Fractions for sharing

start action take this number share them equally of fruit bars. between this number of children.

1. complete the number sentence below, recording what was done and the result.

2. Write a shorter number sentence with the same meaning.

shared between

Num 7.8 Fractions as quotients (cont.)

257

activity 2 Predict, then press [Num 7.8/2]

Anactivityforchildrenworkingintwosorthrees.Itspurposeistoconsolidatetherelation between fractions in bar notation and in place-value notation; and to bring the second of these into the present context of fractions as quotients.

Materials • Activity board, see Figure 27.* • Start cards (numbered from 2 to 10).*† • Action cards (numbered 2, 4, 5, 8,10).*† • Calculator. • Pencil and paper. * Provided in the photomasters †Tofitthesquaresontheactivityboard

What they do 1. A card is put into each of the Start and Action squares. These may be randomly chosen(topofashuffledpile);orchildrenmaybeallowedtochooseiftheywish to experiment systematically.

2. Eachchildthenwritesthequotientfirstinbarnotation,andthenindecimalno-tation. E.g.,

(Thinking: 7 shared between 5 gives 1 each, and 2 over. These 2 shared between 5 give each, as in Activity 1).

3. Their prediction is then tested by using the calculator to give the quotient. 4. The8cardshouldbetakenoutoftheActionpackuntilthechildrenareprofi-

cient with the easier divisors.

Num 7.8 Fractions as quotients (cont.)

÷ =actionstart

Copy the above.Write the quotient in fraction bar notation.Then write it as it would be shown by a calculator.Test your prediction.

Figure 27 Predict, then press

2 5 7 ÷ 5 = 1 and

= 1.4

25

258

activity 3 “are calculators clever?” [Num 7.8/3]

Ateacher-leddiscussionforupto6children.Itspurposeistoencouragethemtobecritical about the results obtained by calculator.

Materials • Calculators.* • Example cards.** * At least one for every 2 children. ** Six are given in Figure 28 and in the photomasters. These should be on separate

cards.

1. 7 people are to be shared between 2 cars, for a journey. How many will there be in each car?

2. 3 children have 2 parents between them. How many parents do they have each?

3. I have 9 carrots, to give to 4 horses. What would be a fair share for each horse?

4. I have 7 potted plants, to give to 4 friends for their birthdays. How many should I give to each?

5. A householder gives 3 children who do some gardening $4.20 to be shared equally among them. How much should they have each?

6. Apackagecontains8fishfingers.Howshouldthesebesharedequallyamongafamilyof 3?

Figure 28 ‘Are calculators clever’ example cards.

What they do 1. Begin with this example. “7 people are to be shared between 2 cars, for a journey. How many will there

be in each car?” 2. Ask them to use their calculators to work out the answer, just to see what

happens. 7 ÷ 2 = 3.5 Whatdotheythinkaboutthis? 3. Ask if they can think of a number story for which this number sentence would

make sense. Here is one. “7 biscuits are to be shared between 2 climbers. How manyshouldtheyhaveeach?Itmakesevenbettersenseifinsteadof3wholebiscuitsand5tenth-partseach,theyget3and1half!Didanyonerememberthat 5 tenth-parts is equivalent to 1 half?

4. Repeat steps 1 and 2 with other number stories. Of the 6 given here, 3 make sense and 3 do not. For those that do not step 3 should be repeated. For those that do, they should invent a number story which does not make sense. (Number 6mayneeddiscussion,andpossiblyeventheintroductionofPlasticinefishfingers!)

Num 7.8 Fractions as quotients (cont.)

259

activity 4 number targets by calculator [Num 7.8/4]

Agameforchildrentoplayintwosorthrees,sharingacalculator.Itspurposeisto increase their knowledge of the relationships between quotients and decimal fractions.

Materials For each group of 2 or 3: • A calculator. • Set of calculator target cards.* • Pencil and paper for each child. * See examples in step 2 below. A full set is provided in the photomasters.

Stage (a) Fractions only 1. Thefractioncardsareshuffled,andputfacedown.(Theunitcardsarenotused

in stage 1.) 2. Inturn,eachplayerturnsoverthetopcardfromthepile.E.g.,

3. He is then allowed 3 ‘shots’ with the calculator. Each shot consists of 4 entries: (number) (÷) (number) (=). 1 point is scored for each shot which hits the target. Inthiscase,somesuccessfulshotswouldbe

4 ÷ 10 8 ÷ 20 2 ÷ 5

4. 3 hits could also be scored by

4 ÷ 10 40 ÷ 100 400 ÷ 1000

For beginners there is nothing against this – they learn something from it. But more experienced players may agree to allow only one shot with a power of 10 as divisor.

5. Itisagoodideaforoneplayertomakehisshotsverbally,andanothertoenterthese on the calculator and show the result.

6. After an agreed number of turns, the winner is the player who has scored most hits.

Stage (b) Mixed numbers 1. Bothpacksofcardsarenowused.Theseareshuffledandputfacedownintwo

piles. 2. Inturn,eachplayerturnsoverthetopcardfromeachpile,andputsthesesideby

side to give a mixed number. There are 72 possible combinations; e.g.,

Whentheonespileisfinished,itisshuffledandusedagain. 3, 4, 5. These are the same as steps 3, 4, 5 in Stage (a).

Num 7.8 Fractions as quotients (cont.)

Rules of the game

.4

2 .35

260

InActivity1,wegobacktoMode1torelatethenewmeaningoffractions,namelyquotients, to a typical physical situation. This helps to show that although the two aspects of division give the same result, this is not something to be taken as obvious from the beginning. This new meaning is of course additional to the other earlier meanings, not a replacement.

Calculatorsarethebestwayfordoingallbutthesimplestdivisions.Itishoweverimportant that they be used with understanding, of two kinds. One kind is relating the answers obtained by calculator to their existing number schema; the other kind is relating these answers critically to situations such as those described in the number stories. These are the purposes of activities 2 and 3 respectively.

Activity 4 is a game for exercising and consolidating their understanding.

OBserVe aND listeN reFleCt DisCuss

Num 7.8 Fractions as quotients (cont.)

Discussion of activities

Number targets by calculator [Num 7.8/4]

261

Num 7.9 rOuNDiNg DeCiMal FraCtiONs iN plaCe Value NOtatiON

Concept Rounding decimal fractions in place value notation to the nearest tenth or hundredth.

Ability To write and say a fractional number as described.

This is not a new concept, but an extension of the concept already learned in Num 2.14 to the present context. The discussion of concepts at the beginning of Num 7.5 is also very relevant here.Followingonfromthis,theword‘decimal’isanadjectivenotanoun.WhenIhearpeopletalkabout‘decimals’Iwanttoask“Decimalwhat?”Theanswerinthepresentcontextisdecimalfractions,andwhilechildrenarelearningIthinkweshould say so.

activity 1 rounding decimal fractions [Num 7.9/1]

Anactivityforasmallgroupofchildren,uptofiveorsix.Itspurposeistoapplytheir existing concept and ability of rounding, acquired in the context of whole num-bers, to fractional numbers in place-value notation. This should be an easy transition providedthattheyhavedoneNum2.14first.Ifthiswassometimeago,itmightbeas well to review this before continuing with the present activity.

Materials • A calculator, for the group. • Pencil and paper for each child.

What they do 1. The player whose turn it is to start has the calculator. She chooses two two-digit prime numbers and divides one by the other. (This is to be sure of generating aninfinitedecimal.)Shereadsoutherresult,andeveryone(includingherself)writes it down. They then continue in much the same way as in Num 2.14/3.

2. Example. The starting player chooses 31 divided by 59, result 0.5254237. . . . They all write this down: 0.5254237

3. Shethensays“Roundedtothenearesttenth-part,thisiszeropointfive.Round-edtothenearesthundredth-part,thisiszeropointfivethree.Roundedtothenearestthousandth-part,thisiszeropointfivetwofive.”Foreachcorrectan-swer,shescoresthreepoints.Ifaplayermakesamistake,shelosestherestofher turn and the next player starts again as in steps 1, 2, 3.

4. The others listen and check. 5 This continues until all have had their turns. A further round may be played if

desired.

Notes (i) To say ‘Oh’ for zero is a mistake, and not less so because it is very common. ‘Oh’ is the name of a letter, not a number.

(ii)Itisalsocorrecttosay(e.g.)“Roundedtotwodecimalplaces.”Thewordinginthe example is used because it also gives practice in remembering what the vari-ous places denote.

(iii)Thisactivitymayofcoursebeextendedasfarasyouthinkfit.However,thou-sandth-parts are as far as it is usually necessary to go.

Discussion of concept

262

activity 2 Fractional number targets [Num 7.9/2]

A game preferably for two or three, though larger numbers are possible, and it may alsobeplayedsolo.ItisadirecttransferofNum6.9intothepresentcontext,andgives plenty of practice in rounding. This also provides an interesting application, sincewithoutroundingitmightsometimesbeverydifficulttoachievethetarget.

Materials For each group: • A calculator. • Three 0 - 9 dice (may be shared with another group). Alternatively, a 0 - 9 pack

ofnumbercardsmaybeshuffledandusedtogivethestartingnumbers. For each player: • Pencil and paper.

1. The three dice are thrown to give a 3-digit number. All the players write this at the top of their paper.

2. Two dice are then thrown to give a 2-digit number. This is preceded by ‘0.’ and written as the target. E.g.,

Dividend Target 549 0.86 3. Theaimistofindanumberwhich,whenusedasdivisor,givesthetargetnumber

as quotient when the calculator result is rounded to the nearest hundredth. Each player in turn writes her attempt, and passes paper, pencil, and calculator to the next player. Example:

Dividend Target 549 0.86

First player: 549 ÷ 800 = 0.68625 which rounds to 0.69(Quitegoodforafirsttrial.)Next player: 549 ÷ 900 = 0.61 —> 0.61 (This player changed the divisor in the wrong direction. A bigger divisor doesn’t give a bigger result!)Next player: 549 ÷ 700 = 0.7842857 —> 0.78(This player learned from the other’s mistake).Next player: 549 ÷ 620 = 0.8854839 —> 0.89Next player: 549 ÷ 630 = 0.8714286 —> 0.87Next player: 549 ÷ 635 = 0.8645669 —> 0.86 . . . the target number.

4. Another round may then be played.

Some calculators allow us to choose how many decimal places they show, but the less expensive ones give us all they can. This is not a disadvantage in a school situ-ation, since it gives good practice in rounding. Activity 1 makes this the focus of the activity rather than incidental to whatever else the calculator is being used for, so thatchildrenbecomefluentinthisskill,andhaveitavailableforusewheneveritisrequired as part of some other task. Activity 2 is a situation of this second kind.

Rules of the game

Discussion of activities

Num 7.9 Rounding decimal fractions in place value notation (cont.)

263

[Space 1] Shape Shapes in the environment and in mathematics

Space 1.8 PARALLEL LINES, PERPENDICULAR LINES

Concepts (i) ‘Is parallel to’ (ii) ‘Is perpendicular to’ as relationships between two lines.

Abilities (i) To recognize examples of parallel or perpendicular lines. (ii) To construct examples of parallel or perpendicular lines.

Just as we have relationships between two numbers (e.g., ‘is greater than,’ ‘is equal to’), so also we have relationships between two lines such as those in the present topic. If line a is parallel to line b, then line b is parallel to line a, so we may also say that these lines are parallel (meaning, parallel to each other). This is true also of the relationship ‘is perpendicular to.’ This reversibility does not hold for all relationships. E.g., it is true for ‘is equal to,’ but not for ‘is greater than.’ Here, however, our main concern is with the particular relationships named in this topic, not with the ways in which relationships themselves may be classified.

activity 1 “My rods are parallel/perpendicular.” [Space 1.8/1] An activity for up to 6 children. Its purpose is to help children learn these two rela-

tionships.

Materials • A pack of 20* cards. On 10 of these is written ‘parallel’ with an example, and on the other 10 is written ‘perpendicular’ with an example.**

• For each child, 2 rods of different lengths. A square section is useful to prevent rolling.

* Any even number of cards will do, provided that there are enough to give a good variety of examples. In the examples, it is important that the pairs of lines should be of different lengths, and oblique relative to the edges of the paper. (See illustrations in ‘Discussion of activities.’) Parallel and perpendicular are relationships between lines, independently of how these lines are positioned on the paper.

** Provided in the photomasters

What they do Stage (a), with cards 1. The cards are shuffled, and put face down. 2. Each child in turn takes a card and puts it in front of him face up. 3. The children then put their rods on top of the lines in the illustration. 4. In turn, they show their cards to the others and say, “My rods are parallel,” or

“My rods are perpendicular,” as the case may be. 5. Each child then takes another card, which he puts face up on top of the card he

has already. Steps 3 and 4 are then repeated. Stage (b), without cards 1. Each child in turn puts his rods either parallel or perpendicular, and says “My

rods are parallel/perpendicular” (as the case may be). 2. The others say “Agree” or “Disagree.” 3. A child may deliberately give a false description if he chooses. The others

should then all disagree.

Discussion of concepts

264

Space 1.8 Parallel lines, perpendicular lines (cont.) activity 2 “all put your rods parallel/perpendicular to the big rod.” [Space 1.8/2] An activity for up to 6 children. Its purpose is to consolidate the concepts parallel

and perpendicular.

Materials • Ruler or big rod. • Small rod for each child (It is good if the small rods are of assorted length.)

What they do 1. The teacher puts down the big rod and says “All put your rods parallel to the big rod.”

2. The children do so. 3. Steps 1 and 2 are repeated several times. Encourage variety in the placing of the

children’s rods. 4. Then, after step 2, the teacher removes the big rod and asks the children what

they notice about their own rods. It should be brought out in discussion that the children’s rods are all parallel to each other.

5. Steps 1, 2, 3, and 4 are repeated with the instruction “. . . perpendicular to the big rod.”

activity 3 Colouring pictures [Space 1.8/3] An activity for 2 to 6 children.

Materials • A picture for each child, made up of lines which are all in parallel or perpendicu-lar pairs. The pictures should be drawn with faint lines. (See examples in the illustration below and in the photomasters.)

• A pack of parallel/perpendicular cards, as used in Activity 1.

What they do 1. The pack is shuffled and put face down. 2. The first child turns over the top card. 3. He is then allowed to colour two lines in his picture, which must be either paral-

lel or perpendicular according to the card. 4. The other children in turn do steps 2 and 3. 5. Putting pencils on top of the lines helps to show up which lines are parallel or

perpendicular. 6. Continue until all the pictures have been coloured.

265

Discussion of activities

Space 1.8 Parallel lines, perpendicular lines (cont.)

Activity 1 is for building the concepts parallel and perpendicular from a variety of examples. Again I emphasize the importance of choosing examples which do not link these concepts either with length of line, or position on paper. If this mistake is not avoided, children will be able to recognize examples like these:

but not like these:

Activity 1 is also for linking the concepts with the appropriate vocabulary. ‘Perpendicular’ is quite a hard word, so you may decide to give children practice in saying it. Initially they are not expected to read these words from the cards. They learn them orally, linked with the visual examples of parallel and perpendicular lines. In this way they will gradually learn to recognize the written words. Activity 2 uses the newly formed concepts to generate examples, with peer-group checking. Activity 3 is similar, but develops a situation in which more than two rods are involved. Of these, a given pair may be either parallel or perpendicular. Activity 4 is the payoff. We now have a picture with many lines. Correct rec-ognition of relationships between these lines allows children to colour their lines. This is a development from an earlier activity in SAIL Volume 1, Space 1.3/1, making use of more advanced mathematical ideas. Why more advanced? Because being straight or curved is a property of a single line; being parallel or perpendicu-lar is a property of a pair of lines – a relational concept.

OBSERVE AND LISTEN REFLECT DISCUSS

266

Space 1.9 CIRCLES

Concepts (i) Circles as shapes. (ii) Diameter, radius, circumference of a circle.

Abilities (i) To recognize circles. (ii) To name the above parts of a circle.

In this topic we focus on the shape aspect of circles. The various interesting rela-tions between circles and other shapes, such as triangles and polygons, will be dealt with in a later topic, Space 1.17, ‘Inter-relations of plane shapes.’ So this is a simple, introductory treatment only, except for Activity 4.

activity 1 Circles in the environment [Space 1.9/1]

A teacher-led discussion for up to 6 children. Its purpose is to teach the concept of a circle as the shape common to a variety of objects in the environment.

Materials • A cutout circle for each child. The circles should be of varying colours and sizes, the smallest being at least 10 cm in diameter.

1. Give each child a circle. 2. Ask if they know the name of this shape. Probably they will: if not, tell them. 3. Ask them to try and find objects in the environment which have this shape.

Some (e.g., a traditional-style clock) will be seen flat-on, others, though in fact circular (e.g., top of a wastepaper container) will be seen foreshortened. These may need to be moved before it is clear to some children that the shape is like that of their cutout model circle.

activity 2 parts of a circle [Space 1.9/2]

A teacher-led discussion for up to 6 children. This is a continuation of Activity 1. Its purpose is to add further detail (interiority) to the concept of a circle.

Materials • The same as for Activity 1, with an additional circle for yourself.

1. Fold your circle in half, and open it out again.

Discussion of concepts

Suggested sequence for the discussion

Suggested sequence for the discussion

267

2. Tell the children to do likewise. 3. Tell them the name for the new line (diameter). Label your diameter, and let

them copy yours. You may like to have a line ruled along the diameter to make it show up more.

4. Fold along another diameter, preferably not perpendicular to the first, and open out again. Put a dot where the creases cross.

5. Ask the children to suggest names for this point. ‘Middle’ is a good name. The mathematical term is ‘centre.’

6. Give them the names for the other parts, and let them label these as before. Only the radius (i.e., half diameter) should be pencilled in.

activity 3 Circles and their parts in the environment [Space 1.9/3]

A teacher-led discussion for up to 6 children. This is a continuation of Activity 1. Its purpose is to consolidate the newly developed aspects of the concept of a circle.

Materials • The labelled circles from Activity 2.

1. Ask the children to find circular objects in the environment where they can also find some of the other parts which they now have labelled in their own circles.

2. A clock of the traditional kind provides a good example. The whole face is a circle; the rim round the edge is its circumference; the spindle of the hands is the centre; and the long hand is most of a radius.

3. A more subtle example is provided by the rim of the kind of wastepaper contain-er often found in school classrooms. This forms the circumference of a circle, but the rest of the circle (its interior) is missing. The bottom of the container has both.

4. Radii and diameters are harder to find. They may be added to available objects (e.g., the wastepaper container again) by stretching strings across. One will pro-vide a diameter. A second string crossing the first will give four radii.

diameterradius

semicircle

Suggested sequence for the discussion

Space 1.9 Circles (cont.)

diameter

circu

mference

268

activity 4 patterns with circles [Space 1.9/4]

An exploratory activity for up to 6 children. Its purposes are to develop skill in drawing, and to provide a ground from which further properties of a circle may be found.

Materials For each child: • Safety compasses. • Paper. • Pencil.

What they do 1. Introduce the children to the use of the safety compasses, and allow them time to practice drawing circles of various sizes. In some of these they should draw and label the centre, a radius, and a diameter, to review and consolidate Activity 2.

2. Each then experiments with making patterns from circles. Some interesting ones can be produced: a few examples are shown in Figure 29. It may be good for them near the beginning to copy these, since there are various things to be found out this way. Straight lines may be added if desired.

3. One at a time, the diagrams are looked at together and discussed. The purpose is to find properties of the circle which are brought into view by the design. For example, in design A each small circle has half the diameter of the big circle. In B, the large circle is divided into two equal parts. (Why?). This pattern may be extended, as in C. In D, the lines joining the centres from an equilateral triangle, and pass through the points where the circles touch. With one more equal touch-ing circle, we get a rhombus. (Why?).

We begin with cutout circles, rather than objects in the environment, in order to provide physical examples with the minimum of other qualities which have to be ignored. The technical term is ‘low-noise examples.’ When the concept of a circle has been formed, it is consolidated by looking for and recognizing examples where there are more distractors – where more abstracting has to be done. This process is repeated in Activities 2 and 3 for those parts of a circle which we want the children to learn at this stage. These 3 activities should follow each other fairly closely. Activity 4 is more advanced, and requires the physical skills involved in draw-ing; so it is good for developing these. Children enjoy making patterns with circles, and there are many important properties which can be found by studying these patterns. At this stage our task is to help children find them and put them into words: geometric proof is something for much later.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 1.9 Circles (cont.)

269

Figure 29 Patterns with circles

a

B

C

D

Space 1.9 Circles (cont.)

270

Space 1.10 COmPARISON OF ANgLES

Concept The shape and size aspects of angle.

Ability To decide which of two angles is the larger/smaller.

When two straight lines meet, they form a shape which we call an angle. The eve-ryday and mathematical meanings are in this case much the same. An angle can also represent a difference between two directions. Here we concentrate on angles as shapes, which may be considered either by themselves, as in this topic, or as contributing to descriptions of other shapes, as in Space 1.14. Objects may be ordered in various ways; e.g., persons by age, names alphabeti-cally, lines by their lengths, sets by their number. Angles may be ordered by their size, which is independent of the length of their arms.

activity 1 “all make an angle like mine.” [Space 1.10/1]

An activity for 2 to 6 children. Its purpose is to introduce the concept of an angle, in its shape aspect.

Materials • An angle disc for every child, preferably each of a different colour. • A hinged angle made from two milk straws and a pipe cleaner. Note An angle disc is made from two circular discs of thin cardboard, one white and

one coloured. Each is cut along one radius, and the two are then interlaced so that one can be rotated relative to the other. In this way a coloured angle is formed which can be adjusted to any desired size.

What they do Stage (a) using angle discs only. 1. On receiving their angle discs, children are allowed a little time to explore. 2. Explain that the coloured shape is called an angle. 3. Child A sets his disc to any angle he likes, puts it in the middle of the table, and

says “All make an angle like mine.” 4. The other children make angles of the same size, and put them near that of child A. 5. Child A then compares them with his own, and pointing to each in turn says

“Agree” or “Disagree.” (The match need only be approximate, and visual com-parison is all that is required at this stage.)

6. Another child becomes child A, and steps 3 to 5 are repeated. Stage (b) using the hinged angle. The steps are the same as in Activity 1, except that child A uses the hinged angle.

This allows physical comparison if there is disagreement.

Discussion of concept

271

activity 2 “Which angle is bigger?” [Space 1.10/2]

An activity for 2 to 6 children. Its purpose is to introduce comparison by size of angles.

Materials • Two angle discs, of different colours. For Stage (a) these should be of the same size, for Stage (b) of different sizes.

• Clear acetate sheet (as used with overhead projectors). • An overhead projector pen (or other non-permanent marker). • Damp rag.

What they do Stage (a) uses discs of the same size. Stage (b) uses discs of different sizes. Otherwise, the stages are alike. 1. Child A and child B each makes an angle. These are put in the middle of the

table. 2. Child C then says (e.g.) “The red angle is bigger than the green angle.” (Or if he

likes, “The green angle is smaller than the red angle.”) 3. In the event that the angles look alike, C would say, “These angles are about the

same size.” 4. The other children say in turn “Agree,” or “Disagree.” 5. If there is disagreement, a check may be made by putting the acetate sheet on top

of one angle and tracing this angle, using an overhead projector pen. 6. Other children take over the roles of A, B, C, and steps 1 to 4 are repeated.

activity 3 Largest angle takes all [Space 1.10/3]

A game for 3 to 5 children, based on Activity 2. Its purpose is to consolidate the concept of comparison by size of angle.

Materials • A pack of cards on which are drawn angles, 6 of each of the following sizes: 30, 60, 90, 120, and 150 degrees.* The arms of the angles should vary considerably in length among angles of the same size: see the ‘Discussion of activities’ fol-lowing Activity 4.

• Paper and pencil for scoring. • Acetate sheet, non-permanent marker, damp rag, as for Activity 2. * Provided in the photomasters

Rules of play 1. The pack is shuffled. Five cards are then dealt to each player. 2. Players put their cards in a pile face down. 3. Starting with the player to the left of the dealer, each player turns over one card

and puts it down face up in front of her. 4. The player who puts down the largest angle takes all the others. She puts this

set of cards in a pile in front of her. Comparison may be visual, or by the acetate sheet method as in Activity 2.

5. If two angles are equal in size, each player takes one angle for her pile.

Space 1.10 Comparison of angles (cont.)

272

6. The winner of a round is the first to put down a card for the next round. The oth-ers follow in turn, clockwise.

7. When all the cards have been played, the players’ scores are recorded according to the number of cards they have taken.

8. Another game may then be played, and the scores added to those of the previous game.

9. This game may also be played as ‘Smallest angle takes all.’

activity 4 angles in the environment [Space 1.10/4]

An activity for relating the concept of angle with examples in the environment.

Materials • An angle disc for each child, as in Activity 1.

What they do 1. Each child looks around the room and finds an angle. (Suitable sources are angles made by hanging strings, angles between books on shelves, corners of tables, hands of a clock. Many environmental angles are right angles. This does not matter: it prepares the way for the next topic.)

2. Each child adjusts his angle to the same size as whatever he has chosen. 3. In turn, each child holds up his angle and says (e.g.) “I’ve made my angle the

same size as the angle of that string” (pointing). 4. The others in turn say whether they agree.

Activity 1 is for forming the concept of an angle, by making and copying a variety of angles of different sizes and colours. A different example of the same concept is introduced in stage 2, in the form of a hinged angle made from milk straws and a pipe cleaner. Once the concept of an angle has been established children are in a position to compare angles of different sizes. Note that the size of an angle has nothing to do with the length of its arms. Of the two angles below, the left is the larger. This is the point of the second stage of Activity 2, and of Activity 3.

Activity 4 relates the concept of angle to environmental examples. Here there is more irrelevant detail to be ignored, so it comes last, after the concept has been established by simpler examples.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 1.10 Comparison of angles (cont.)

273

Discussion of concepts

Space 1.11 CLASSIFICATION OF ANgLES

Concepts (i) Acute angle, obtuse angle, right angle. (ii) (Later) Straight angle, reflex angle.

Abilities (i) To classify angles into these categories. (ii) To produce examples of these categories.

These are simple concepts, once children can compare angles by size. An acute (sharp) angle is one which is less than a right angle, an obtuse (blunt) angle is one which is greater than a right angle. Among obtuse angles, we do not usually include angles greater than a straight angle, which is two right angles put together. These are called reflex angles.

activity 1 Right angles, acute angles, obtuse angles [Space 1.11/1]

A sorting activity for 1 to 3 children. Its purpose is to help children form the con-cepts of acute angle , obtuse angle, right angle.

Materials • A piece of paper for each child. (The size is not critical.) • Coloured crayon or felt tip pen. • An assortment of cutout angles, in the form of sectors of a circle. (See illustra-

tion below). There should be roughly equal numbers of acute, obtuse, and right angles. The circles should be of assorted sizes; assorted colours too, if you like. A suitable set is provided in the photomasters.

• 3 set loops. • 3 small cards labelled ‘Acute angles,’ ‘Obtuse angles,’ ‘Right angles,’ respec-

tively.

What they do 1. Each child makes herself a right angle by folding her paper twice.

Sample cutout angles

first fold second fold

274

2. The right angle (at the double fold) is then marked with coloured crayon or felt tip.

3. The cut out angles are mixed together in the middle of the table. 4. The 3 set loops are put out, with a card labelling each: respectively acute angles,

obtuse angles, right angles. 5. The children work together to sort these into 3 sets: acute, obtuse, right angle.

They do this by comparing each with their right angles. 6. A good way is for each child in turn to put an angle into a set loop, saying as she

does so (e.g.) “Obtuse angle.”

activity 2 angle dominoes [Space 1.11/2]

A game for 2 to 4 players. Its purpose is to consolidate the concepts formed in Activity 1.

Materials • A set of 24 angle dominoes. These are of two kinds (see Figure 30). A full set is provided in the photomasters. Note that the right angles are not marked as such, so that the players have to decide for themselves, either by observation or by comparing with their own right angles.

Rules of play 1. The rules are the same as for ordinary dominoes. There are, however, various ways of playing. A way we find goes well is described in steps 2 to 5.

2. 5 dominoes are dealt to each player. The rest are put face down in the middle. 3. A player who cannot go takes a domino from the middle. This counts as her

turn. 4. When all are taken from the middle, a player who cannot go knocks once. If she

cannot go a second time she knocks twice; if a third time, she knocks thrice and is out of the game.

5. The winner is the first to use all her dominoes; or the one left with fewest, if none can go.

6. In the present game, the match must be between an angle and a word, or be-tween two angles of the same kind (acute, obtuse, right). Two words may not be matched, since this does not depend on understanding their meaning.

Space 1.11 Classification of angles (cont.)

rightangle

rightangle

Figure 30 Angle dominoes

275

activity 3 “Mine is the different kind.” [Space 1.11/3]

A game for exactly 3 players.

Materials • 30 cards, on 10 of which are drawn acute angles, on 10 right angles, and on 10 obtuse angles. As provided in the photomasters, the arms of the angles on the cards should vary considerably in length.

• They also keep their right angles from Activity 1, to check.

Rules of play 1. The cards are shuffled, and all are then dealt to the players. 2. The players hold their packs face down. 3. Starting with the player to the left of the dealer, each in turn puts down his top

card face up in front of him. 4. If all are alike or all are different, the players continue in turn to put down anoth-

er card on top of their earlier one(s), until there are two alike and one different. Here ‘alike’ refers to the 3 categories acute, obtuse, right angle.

5. The player whose pile shows the odd one out says “Mine is the different kind.” or “Mine is the odd one out.” and explains why. E.g., “Yours are acute, and mine is obtuse.” He then takes all 3 piles, which he puts face down at the bot-tom of his own pile.

6. This player then puts down a card, and steps 3, 4, 5 are repeated. 7. If a player has no more cards in his hand to put down, his pile stays as it is until

step 5 applies. 8. Play continues until one player has lost all his cards. The winner is then the one

with most cards. 9. If this takes too long, the winner is the player who finishes with most cards.

activity 4 “Can’t cross, will fit, must cross.” [Space 1.11/4]

An activity for up to 4 children.

Materials • An assortment of cutout angles.* • 12 instruction cards.** On 4 of these is written “Each of you take an acute angle”; on

4, “Each of you take a right angle”; and on four, “Each of you take an obtuse angle.” • One angle card as illustrated in step 2.** * The same as for Activity 1 ** Provided in the photomasters

What they do Stage (a) 1. The cutout angles are mixed together and spread out in the middle of the table. 2. The angle card is put on the table with the Side 1 showing.

Space 1.11 Classification of angles (cont.)

Side 1

Put two angles together with their points on the dot, on the same side of the line if you can.

276

3. The instruction cards are shuffled and put face down. 4. The top instruction card is turned over, and the children do as it says. 5. They then follow the instruction on Side 1 of the angle card. 6. Steps 4 and 5 are repeated until all the instruction cards are used.

Stage (b) 1. Steps 1 to 6 are the same as in Stage (a), except that Side 2 of the angle card is

used. Examples of correct predictions are shown below.

“Can’t cross” (2 acute angles)

“Will fit.” (2 right angles)

“Must cross.” (2 obtuse angles)

7. The intention is that children discover the basis for prediction, as shown above in the three parentheses.

Space 1.11 Classification of angles (cont.)

Side 2

Predict whether two angles of the kind you have can’t cross, will fit, or must cross. Then test on this card.

277

8. When they have done so, they may be invited to put this into words. 9. They may also be asked if they can think out why this is so. This, however, is

difficult for children of this age.

Activity 1 is for concept building from physical examples (Mode 1). Activity 2 is for consolidating the link between words and concepts, using a variety of exam-ples. Testing in this case is by agreement and, if necessary, discussion (Mode 2). Activity 3 consolidates these concepts and their associated words in another game, involving finding two of a kind and one of a different kind. Activity 4 investigates some further consequences of the properties, and sets children on the path of mak-ing mathematical generalizations.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 1.11 Classification of angles (cont.)

Discussion of activities

“Can’t cross, will fit, must cross.” [Space 1.11/4]

278

Space 1.12 CLASSIFICATION OF POLygONS

Concepts (i) That of a polygon. (ii) Ways in which polygons may be classified: (a) regular and non-regular (b) number of sides.

Note Varieties of triangles and quadrilaterals are dealt with in later topics.

Abilities (i) To classify polygons in the ways described above. (ii) To name them, and state their characteristic properties.

In this topic we make a start with thinking about the various kinds of shapes which can be made from straight lines. These are called polygons. We look for ways of classifying them, and start by using regular/irregular, and number of sides. In subsequent topics, further classifications within the set of triangles and the set of quadrilaterals will be introduced.

activity 1 Classifying polygons [Space 1.12/1]

A teacher-led discussion for up to 6 children. Its purpose is to help children to dif-ferentiate among the various kinds of polygon.

Materials • Polygons, pack 1.* This consists of 30 square cards, on which are 15 regu-lar polygons and 15 non-regular polygons. The regular polygons consist of 3 equilateral triangles, 3 squares, 3 pentagons (5 sides), 3 hexagons (6 sides), 3 octagons. The 15 non-regular polygons are as above, except that no two sides or angles are of equal size. (So the triangle is scalene, and the squares are replaced by non-regular quadrilaterals.) The sizes of the polygons should be varied: but there should not be a clear large/small distinction.

* Provided in the photomasters

1. Lay out an assortment of the polygon cards, about half the pack. 2. Ask for suggestions how these might be sorted. The two ways we have in mind

are (a) number of sides, and (b) regular and irregular. These we hope are the most noticeable.

3. If other ways of sorting are proposed, give them a fair hearing and let children try to sort their way. Size is quite a reasonable one, but we have tried not to embody this in the material.

4. Eventually the children will arrive at (a) and (b). Usually (a) is noticed first. 5. Put down the rest of the pack, and let each child collect the set of all shapes

which have a particular number of sides. 6. When they have done this, let each describe in his own words the set he has col-

lected. E.g., “I have collected the set of shapes with 5 sides.”

Discussion of concepts

Suggested sequence for the discussion

279

7. Then tell them the mathematical names for these, and let them all describe their sets again.

8. Ask them to look at their sets, and see if they can sort them further. Usually they will now see the regular /irregular distinction.

9. When they have sorted into these subsets and described in their own terms, introduce them to the generally used words. E.g., “The set of regular hexagons, and the set of non-regular hexagons.”

10. Mix all the pieces, and repeat steps 5 to 9 with children collecting different sets.

activity 2 polygon dominoes [Space 1.12/2]

A game for 2 to 4 players. Its purpose is to consolidate the concepts formed in Ac-tivity 1, and link them with their names.

Materials • A set of dominoes, as provided in the photomasters.

What they do This game is played according to the usual rules of the game of dominoes, except that a match of word with word is not allowed. (See Space 1.11/2, ‘Angle domi-noes,’ for these rules.) The following are some further suggestions which may be incorporated if and as desired.

1. It is good for each player to have a mixture of word dominoes and figure domi-noes. To achieve this, first sort the cards into words and pictures. Then deal these two packs in succession.

2. If they want to score, the first to go ‘out’ scores 5, the next 4, etc. Players who do not go ‘out’ score zero.

3. When learning, it is good to help each other. Later, ‘No help’ may be agreed. Players can then often help themselves by looking at existing matches of words with polygons.

activity 3 Match and mix: polygons [Space 1.12/3]

A game for 2 to 5 players. Its purpose is to consolidate further the concepts formed in Activity 1.

Materials • Polygons, pack 2.* This is a double pack of the regular polygons as in Activity 1, so there will be 30 cards, now including 6 of each kind of regular polygon. It might be helpful to have this pack of a different colour from pack 1.

• A card as illustrated below. * Provided in the photomasters

Space 1.12 Classification of polygons (cont.)

MATCH IX

280

1. The cards are spread out face downwards in the middle of the table. 2. The MATCH and MIX card is put wherever convenient. 3. Each player takes 5 cards (if 2 players only, they take 7 cards each). Alterna-

tively the cards may be dealt in the usual way. 4. They collect their cards in a pile face downwards. 5. Players in turn look at the top card in their piles, and put cards down next to

cards already there (after the first) according to the following three rules. (i) Cards must match or be different, according as they are put next to each

other in the ‘match’ or ‘mix’ directions. (ii) Not more than 3 cards may be put together in either direction. (iii) There may not be two 3’s next to each other. (‘Match’ or ‘mix’ refers to the kind of polygon). Examples (using A, B, C . . . for different kinds of polygon.) A typical arrangement:

None of these is allowed:

A card which cannot be played is replaced at the bottom of the pile. 6. Scoring is as follows. 1 point for completing a row or column of three. 2 points for putting down a card which simultaneously matches one way and

mixes the other way. 2 points for being the first ‘out.’ 1 point for being the second ‘out.’ (So it is possible to score up to 6 points in a single turn.) 7. Play continues until all players have put down all their cards. 8. Another round is then played.

C a a aB B B D DD

aBCD

BBBB CCCDDD

aaBBCC

aBa

Space 1.12 Classification of polygons (cont.)

B a

281

Sorting and classifying are among the most fundamental ways in which intelli-gence functions, and activities based on these have been in use from the very be-ginning. Activity 1 is a sorting activity, leading to two of the most noticeable ways in which polygons can be classified. Once these categories have been learned, they are consolidated by the games in Activities 2 and 3. The cards used for sorting and classifying polygons have two advantages over the printed page. First, they show the figures in a variety of positions, since (being square) they may be any way up. This helps to establish that a figure is the same whatever its position, and accustoms children to recognizing figures in all posi-tions. Second, by allowing physical sorting, it leads children to form the categories for themselves rather than the usual method of presenting them with the finished article. Once again we use a combination of Modes 1 (physical experience) and 2 (dis-cussion, communication of vocabulary by teacher) for schema building, followed by using the newly formed concepts in the co-operative situation provided by a game.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 1.12 Classification of polygons (cont.)

Discussion of activities

Match and mix: polygons [Space 1.12/3]

282

SPACE 1.13 POLygONS: CONgRUENCE AND SImILARITy

Concepts (i) Congruence (ii) Similarity (iii) Proportionality, in the context of similar polygons. Abilities (i) To recognize when polygons are congruent or similar. (ii) To identify and label corresponding vertices and sides. (iii) To use the proportional property of similar polygons to calculate lengths of

sides.

At an everyday level, we find it easy to perceive whether objects are exactly alike, or the same shape but different sizes. Congruence and similarity are the mathemat-ical equivalents of these, and the concepts are easily formed from suitable exam-ples. In the present topic, these examples are all polygons. In mathematics we first go a little further, by noting which points and lines cor-respond to each other in the two figures. But the importance of similarity comes from its property that the ratios of the lengths of corresponding sides are all in the same proportion, and this applies also to the diagonals. It is this property of similar figures which is the foundation of all scale drawings, which includes maps (other than sketch maps). (It might be worth taking a few moments to consider how many areas of everyday and professional life are made easier by these.) So I thought we should give the children a preview. I write ‘preview,’ because a full understanding of this property depends on an understanding of ratio and proportion; and to prove that it is always true (as against reasonable conjecture) depends on geometrical theorems which they will not learn until later. So I have tried to present this important property of similar figures in a way which, though necessarily incomplete, is accurate. This was in fact more difficult to write than if I had first included the topics just mentioned!

Discussion of concepts

283

activity 1 Congruent and similar polygons [Space 1.13/1]

An activity for children to do individually, with guidance. Its purpose is to introduce the first two of the concepts listed for this topic, and help children to acquire the first two of the abilities.

Materials For each child • A paper copy of Photomaster 193 from the Volume 2a photomasters, for use as a

worksheet.* • A transparent copy of the same. (Helpful but not essential.) • Pencil and paper. * Alternatively, a laminated copy may be used with water-soluble pens.

What they do 1. First, they spend a little time exploring the worksheet, to find some pairs of poly-gons which look alike.

2. Are all the pairs alike in the same way? (No. Some are exactly alike except for position. Others are the same shape, but different sizes. Their angles are equal, but their sides are not. The transparent copy is useful for checking this.)

3. Pairs of the first kind are called congruent. Pairs of the second kind are called similar.

4. To show which figures belong to the same pair, they are lettered alike, with suf-fixes 1 and 2 for the first and second of the pair. In the example below, a pair of congruent triangles and a pair of similar quadrilaterals are shown in this way.

5. Notice that the letters are positioned so that they also show which vertices (cor-ners) correspond, i.e., belong together. If this is done correctly, we can also use them to show which sides belong together. E.g., in the above pair of congruent triangles, the sides A1B1 ↔ A2B2 and B1C1 ↔ B2C2, etc.

6. Finally, they compare their results and discuss any points arising.

Space 1.13 Polygons: congruence and similarity (cont.)

A1

A2

B1

C1

B2

C2

E1

F1

D1

G1

D2

F2

G2

E2

284

Space 1.13 Polygons: congruence and similarity (cont.)

activity 2 Sides of similar polygons [Space 1.13/2]

A teacher-led discussion in which children work individually on their own work-sheets. Its purpose is to introduce the concept of proportionality.

Materials For each child: • A copy of the worksheet (SAIL Volume 2a, Photomaster 194).* • Ruler graduated in centimetres and millimetres. • Pencil and eraser. * Provided in the photomasters

What they do 1. They begin with the first pair of polygons, which are similar quadrilaterals, and letter them in the same way as in Activity 1.

2. Here we are interested in the correspondences between the sides, and especially between their lengths. So they measure all the sides of both quadrilaterals, and write these in their worksheets.

3. What do they notice? (Each side of the second quadrilateral is twice the length of the corresponding side of the first.)

They record this as follows.

x 2 A1B1 > A2B2

x 2 B1C1 > B2C2

x 2 C1D1 > C2D2

x 2 D1 A1 > D2 A2

4. They should now repeat steps 1, 2, 3 for pair 2, which is a pair of similar trian-

gles. Does the same relationship apply? (Yes, except that now each side in the larger triangle is three times the size of that in the smaller.)

5. Now for the third pair of polygons, which are non-regular pentagons. They should measure and write in the lengths of all the sides of the smaller, and just one side of the larger — whichever they choose. Then, to think about whether they are ready to make an intelligent guess about the other lengths, but not to speak it until all have had time to think.

6. Any suggestions? We hope that at least one will say something like “In this pair, the multiplier is one and a half for all the sides.” (Meaning the multiplier from the smaller to the larger.) If not, they should measure another side of the larger polygon, and see if this gives them an idea. Likewise if necessary for a third side, by which time light should dawn! (If it does not, then they are probably not ready to continue with this topic and should return to it later.)

285

7. They should then write in one of their predicted lengths for one of the sides of the larger polygon, check by measurement, and, if confirmed, put a check mark against it; if wrong, a cross. Repeat for the other sides, learning, if necessary, as they go along.

8. Does this principle apply to all polygons? (We can’t be certain, but it seems likely. Later on they will be able to prove that is does apply to all polygons.)

9. A general formulation would be appropriate at this stage, but is difficult without the language of proportionality. I would be content with something like the fol-lowing: “For similar polygons, the multiplier is the same for all pairs of corre-sponding sides. So if we know it for one side, we know it for all of them.”

Notes (i) (Step 5.) The mathematician’s word for an intelligent guess is ‘conjecture.’ Some children may like to add this to their vocabulary.

(ii) The multiplier is not always a whole number. (E. g. the last pair.) (iii) At some stage the question may arise whether the same principle applies to the

areas of similar polygons. This is harder to answer. One way would be to ap-ply the grid method of Meas 2.1/4 to find out. Alternatively, we could take two similar squares, one with the sides twice as long as the other. Here the answer becomes obvious — no, it isn’t. In the case of the squares, the area of the sec-ond is four times that of the first, which opens up an interesting line of investiga-tion for some of the children to follow, perhaps, on another occasion. If any-one suspects that the multiplier for the areas is the square of the multiplier for lengths, congratulations. They are quite right. If the question is not raised by any of the children, I suggest that at this stage it may be better not to introduce this additional aspect.

activity 3 Calculating lengths from similarities [Space 1.13/3]

An activity for children to do individually. Its purpose is to develop the third of the abilities listed at the beginning of this topic, namely to be able to use the proportional property of similar polygons to calculate lengths of sides of polygons.

Materials For each child: • A worksheet, see Figure 31.* • Pencil and eraser. • Ruler graduated in centimetres and millimetres. * A copy is provided in the photomasters.

What they do 1. A full size illustration of the worksheet appears on the next page. This includes instructions for the children to follow. I hope that they will find the quest satisfying.

Activity 1 introduces the concepts of congruence and similarity at a perceptual level, and also introduces a notation which calls attention to the related concept of corresponding vertices and sides. Activity 2 develops the concept of proportionali-ty from examples in which this shows clearly from the measurements; and Activity 3 consolidates this concept by using proportionality repeatedly to calculate many lengths from a few given measurements.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 1.13 Polygons: congruence and similarity (cont.)

Discussion of activities

286

Space 1.13 Polygons: congruence and similarity (cont.)

This figure is made up of regular hexagons, parallelograms (including some rhombi), and non-regular pentagons. Some of the measurements are also given. See if you can use your knowledge of congruence and similarity to CALCULATE all the other measurements and write them in. The dotted line is there as a hint. If a figure looks as if it is symmetrical, you may assume that it is. When you have finished, check your results by measuring and/or comparing with someone else.

Figure 31 Worksheet for ‘Calculating lengths from similarities’ [Space 1.13/3]

All the measurements are in centimetres and show the lengths of the line segments between the nearest points of intersection.

287

Discussion of concepts

Suggested sequence for discussion

Space 1.14 TRIANgLES: CLASSIFICATION, CONgRUENCE, SImILARITy

Concepts (i) Ways in which triangles may be further classified: equilateral, isosceles, scalene, right angled.

(ii) Congruence of triangles. (iii) Similarity of triangles.

Abilities (i) To classify triangles in the ways described above. (ii) To recognize whether two triangles are congruent or not. (iii) To recognize whether two triangles are similar, and say in what ways they are

the same and in what ways different.

(i) An equilateral triangle has all its sides equal in length, and all its angles equal in size. An isosceles triangle has two (but not three) equal sides and two equal angles. A scalene triangle has no two sides equal, and no two angles equal. A right-angled triangle has one right angle. Further categories are possible, e.g., obtuse-angled; but the above are sufficient for the present. The ‘right angled’ and ‘isosceles’ categories overlap, and so do ‘right-angled’ and ‘scalene.’(ii) Two triangles are congruent if one fits exactly on top of the other, turning it over if necessary.(iii) Two triangles are similar if they are the same shape but not necessarily the same size. (So congruent triangles are also similar, but would usually be called ‘congruent.’)

activity 1 Classifying triangles [Space 1.14/1]

A teacher led discussion for up to 6 children. Its purpose is to help children form group (i) of the concepts above.

Materials • Triangle pack 1.* This is like the polygons pack used for Activity 1 in the pre-vious topic, but consists only of triangles. These are of 5 varieties, 6 assorted examples of each. The varieties are:

Equilateral Isosceles, right-angled Isosceles, not right-angled Scalene, not right-angled Right-angled, scalene • Set loops. * Provided in the photomasters

This follows the same lines as Space 1.12/1, ‘Classification of polygons.’ There is a difference, in that the categories right angled and isosceles overlap. After children have discovered this for themselves, it may be shown using set loops. Start with the loops not overlapping and let children discover the need to overlap them. The same applies to right angled and scalene.

At this stage, informal descriptions are satisfactory. Children may then be told the mathematical names in preparation for Activity 2.

288

Space 1.14 Triangles: classification, congruence, similarity (cont.)

activity 2 Triangle dominoes [Space 1.14/2]

A game for 2 to 4 players. Its purpose is to help children attach the mathematical names to the concepts formed in Activity 1, while also consolidating the concepts.

Materials • A set of dominoes as provided in the photomasters.

What they do This game is played as in Space 1.11/2 and Space 1.12/2. Reminder: each match must be between a word and a triangle.

activity 3 Match and mix: triangles [Space 1.14/3]

A game for 2 to 5 players. Its purpose is to combine and practice further the con-cepts formed in Activities 1 and 2.

Materials • Triangles pack 1. This is the same as used in Activity 1. • Match and Mix card as in Space 1.12/3.

The game is played in the same way as Space 1.12/3, ‘Match and mix: polygons.’ An interesting development is provided by the overlapping categories. These may be used for either matching or mixing, provided that the player states his reason. E.g.,

“Match:both isosceles.”

“Mix:one right-angledthe other not.”

Rules of the game

MATCH IX

289

Space 1.14 Triangles: classification, congruence, similarity (cont.)

activity 4 Congruent and similar triangles [Space 1.14/4]

A teacher-led discussion for up to 6 children. Its purpose is to introduce these con-cepts, with some of their further properties.

Materials • Triangles pack 2.* This consists of 24 cutout triangles in 12 pairs. Of these, 6 are pairs of congruent triangles, and 6 are pairs of similar triangles. Most of these should be scalene, but 2 pairs in each set could be of one of the ‘special’ kinds (equilateral, isosceles, right angled). Sizes should be assorted. In 2 of the similar pairs, the larger should have sides twice as long as the smaller; in 2 simi-lar pairs, three times as long; and in 2 similar pairs, one-and-a-half times as long.

* A suitable set of triangles is provided in the photomasters.

1. Put all the triangles in the middle of the table and ask children to find pairs of triangles which are alike. Depending on the number of children, each will get 2 or 3 pairs.

2. Ask them in turn to describe in their own words the ways in which the trian-gles in each pair are alike. E.g., (for congruent triangles) “They are exactly the same.” “They fit exactly.” (For similar triangles) “They are the same shape but different sizes.”

3. Tell them the mathematical names for these relationships, and let them in turn describe their pairs again. (E.g., “I have a pair of congruent triangles and a pair of similar triangles.”)

4. Ask them what else they can find out about their pairs of similar triangles.

Among the more important properties are

(i) If you put the smaller on top of the larger like this, with two sides coinciding, the third sides are parallel.

(ii) This can be done in 3 ways. Here are the other two.

Suggested sequence for discussion

290

Space 1.14 Triangles: classification, congruence, similarity (cont.)

(iii) This is easier to follow in pictures than in words. E.g., (measurements in cm)

In mathematical language, the ratios of the lengths of corresponding sides are in the same proportion. This formulation lies several years into the future for the children. At the present stage, any kind of description which shows an intuitive grasp is acceptable.

These activities parallel those in Space 1.12, so it may be worth rereading the discussion at the end of that topic. The advantage of the present approach over the printed page shows particularly clearly in Activity 4, step 4(ii), when the smaller triangle can be physically fitted onto the larger triangle in 3 different ways.

OBSERVE AND LISTEN REFLECT DISCUSS

7 14

3 56 10

Discussion of activities

Congruent and similar triangles [Space 1.14/4]

291

Discussion of concepts

Space 1.15 CLASSIFICATION OF qUADRILATERALS

Concepts Ways in which quadrilaterals may be classified: (i) kite (ii) trapezium (iii) parallelogram (iv) rhombus (v) rectangle (vi) square (vii) oblong

Abilities (i) To classify quadrilaterals in the ways described above. (ii) To name them, and state their characteristic properties. (iii) To relate these classifications.

In Space 1.12, we classified polygons by two criteria: number of sides, and regular/irregular. Then in Space 1.14, we took one of these categories, 3 sided polygons (triangles) and made further distinctions between different kinds of triangle. Now we do the same for 4 sided polygons (quadrilaterals). These we sort by 2 criteria:(a) how many sides are equal, and which;(b) how many pairs of sides are parallel.These are not independent. E.g., if a quadrilateral has two pairs of parallel sides, then these pairs of opposite sides must also be equal. This results in interesting relationships between categories: e.g., a square is also a rhombus, a rectangle, and a parallelogram. Squares and oblongs, as we shall use the terms, are both kinds of rectangle. One sometimes hears people talk about squares and rectangles as though these were distinct categories, but this is rather like talking about girls and children. Girls and boys are both kinds of children. Likewise squares and oblongs both satisfy the criteria for rectangles, which is to have 4 right angles. The categories are described and illustrated below.

Kite 2 pairs of adjacent sides equal, but not 4 equal sides.

Trapezium 1 pair (only) of opposite sides parallel.

292

Space 1.15 Classification of quadrilaterals (cont.)

Parallelogram 2 pairs of sides parallel. (Also, 2 pairs of opposite sides equal)

Rhombus All 4 sides equal. (It is also a parallelogram.)

Rectangle 4 right angles. (It is also a parallelogram.)

Square 4 right angles. 4 equal sides. (it is also a rectangle, a rhombus, and a parallelogram.)

Oblong A rectangle which is not a square.

activity 1 Classifying quadrilaterals [Space 1.15/1]

A teacher-led discussion for up to 6 children. Its purpose is to teach children the above categories and their names.

Materials • Quadrilaterals pack.* This consists of 36 cards, with 6 examples each of cat-egories (i) to (iv) as listed above, 6 squares, and 6 oblongs. (So there will in fact be 12 rectangles: 6 square rectangles and 6 oblong rectangles.) The examples should vary in size, shape (except squares), and orientation. E.g., the rectangles should not all be drawn with their sides parallel to the sides of the card.

• Notebook or sheet of paper for each child. • Pencils. * Provided in the photomasters

What they do 1. Lay out an assortment of the quadrilaterals cards, about half the pack. 2. Explain that the object is to sort these. 3. Invite the children to look for shapes which belong together. Accept any reason-

able suggestions, and let each child collect all the examples of that shape. 4. Put down the rest of the pack, and let them continue until all are sorted. 5. Let each child describe in his own words what he has collected, and then tell him

the mathematical term for this. 6. If any cards remain unsorted, sort these too. What happens now will depend on

the categories which have been used.

or

293

Suggested sequence for discussion

Space 1.15 Classification of quadrilaterals (cont.)

7. If parallelograms and rhombi have been put together, ask if they can be sorted further.

8. If squares and oblongs have been separated, leave them so since this will be dealt with in Activity 2.

9. By now the categories are likely to be fairly near the ones illustrated. If not quite the same, explain that there are various reasonable ways of sorting. The way you are going to show them is a useful one which is widely used, and not very different from theirs.

10. The one to aim at in the present activity is, I suggest, the one shown above with oblongs and squares, parallelograms and rhombi, all separated.

11. Tell them the mathematical names for these, if they do not know already. At this stage, oblongs may be called rectangles (which they are), or oblongs.

12. Each child describes his set using its mathematical name. E.g., “I have collected a set of squares.”

activity 2 Relations between quadrilaterals [Space 1.15/2]

A continuation of Activity 1. Its purpose is to help children find the relationships between the various categories of rectangles.

Materials The same as for Activity 1.

1. Tell the children to keep 2 examples of each of their categories, and collect the rest.

2. Ask for all the parallelograms. 3. Depending on what you have been given, ask about other examples which you

have not been given. E.g., (pointing to the oblongs) “Aren’t those parallelo-grams too?”

4. Discuss what properties a shape must have to be a parallelogram, and get them to formulate the outcome. E.g., “Rectangles are a kind of parallelogram.”

5. Repeat steps 2, 3, 4 until the children have come to realize that squares, oblongs, and rhombi are all ‘kinds of parallelogram.’

6. Repeat as in steps 2 to 5 (a) by asking for all the rectangles. You should get squares and oblongs. (b) by asking for all the rhombi. You should get squares and other kinds of

rhombi. In each case discuss the criteria, and get the children to formulate the outcome.

E.g., “Squares and oblongs are both kinds of rectangle.”

294

Rules of the game

Space 1.15 Classification of quadrilaterals (cont.)

activity 3 “and what else is this?” [Space 1.15/3]

A game for up to 6 children. Its purpose is to consolidate the shape concepts, and relations between them, learned in Activities 1 and 2.

Materials • Quadrilaterals pack. The smaller set with only 2 examples of each kind, used in Activity 2, is more convenient, but the full pack may be used.

1. The cards are spread out face up on the table in such a way that it is easy to find an example of whichever kind is wanted.

2. Then player 1 picks up a card and asks player 2 (on his left) “What is this?” 3. Player 2 might reply “An oblong.” 4. Player 1 asks player 3 (the next player after player 2) “And what else is this?” 5. Player 3 might reply “A rectangle.” 6. Player 1 repeats the question to players 4, 5 . . . in turn. Replies might be “A

quadrilateral,” “A polygon with 4 sides” and eventually “That’s all.” 7. Steps 2 to 6 are then repeated with the next player acting as player 1.

activity 4 “I think you mean . . .” [Space 1.15/4]

A game for 4 or 6 children. Its purpose is further to exercise the shape concepts and relations learned in this topic, with emphasis on language and vocabulary.

Materials • Quadrilaterals pack, as used in Activity 1 but without kites or trapezia. This leaves 24 cards, all parallelograms of one kind or another.

• Card listing the pairs which may be formed: squares, oblongs, rhombi, and par-allelograms which are none of the foregoing.*

* Provided in the photomasters

Rules of play 1. The cards are shuffled and dealt. 2. Each player looks at his cards. 3. The object is to collect pairs of the same kind: squares, oblongs, rhombi which

are not squares, and parallelograms which are none of the foregoing. 4. Any pairs which players already have in their hands are put down. 5. They then try to acquire more pairs by asking in turn, starting with the player on

the left of the dealer. 6. They may ask whom they like, but must not ask for the card by name. So a

player wanting, say, a rhombus might say “Please may I have a parallelogram with all its sides equal?” The one asked might reply “I think you mean a rhom-bus,” followed by “Here you are” or “Sorry.”

7. A player asking by name loses his turn. 8. When a player makes a pair, he puts it down. 9. Scoring is as follows. 2 points for first ‘out,’ 1 point for second ‘out,’ plus 1

point for each pair. 10. Another round may then be played. Note Rule 6 requires that a parallelogram which is not a square, oblong, or rhombus, must

not be called a parallelogram directly. It might be called “A quadrilateral with two pairs of parallel sides, not all equal.”

295

Discussion of activities

Space 1.15 Classification of quadrilaterals (cont.)

Activity 1 is concerned with sorting and classifying – fundamental activities of intelligence which have been in frequent use from the beginning. Activities 2, 3, 4 are more sophisticated, however, since the present classes overlap. This gives the opportunity for reflection on the way these categories are related to each other. Activity 3 starts with an example, and asks in what ways it can be categorized. Activity 4 starts with a category, and ends with a particular member of that category (if the asker is lucky). It also emphasizes the language of description and of relationships.

OBSERVE AND LISTEN REFLECT DISCUSS

“I think you mean . . .” [Space 1.15/4]

296

Suggested outline for the discussion

Space 1.16 CLASSIFICATION OF gEOmETRIC SOLIDS

Concepts The defining characteristics of the better known geometric solids. I suggest the fol-lowing as a basic list: cube, cuboid, prism (at least two kinds), pyramid, and cone. Your set may also include others, such as a tetrahedron.

Ability To state and discuss these accurately, first with and then without the help of physi-cal models. (We are not, however, expecting definitions which are mathematically rigorous.)

They should be familiar with the concepts and names of the solids themselves, or at least most of them, having been introduced to them in Space 1.7 in Volume 1. We are now examining these in more detail, and moving towards a statement of defining characteristics for each.

activity 1 What must it have to be . . . ? [Space 1.16/1]

A teacher-led discussion for a small group. Its purpose is to help them arrive for themselves at the defining characteristics of the solids listed above.

Materials • A set of geometric solids.

1. A cube is an easy example to start with. Put one out for them to look at, and ask what it is called. (A cube.)

2. “What must it have, to be a cube and nothing else?” (A cube has six faces, all squares. It is probably enough to say that all of its faces are squares, since it is difficult to think of a solid of which the faces are all squares which is not a cube. But it might not be easy to prove that this is necessarily so.)

3. Next, I suggest, a cuboid. “What must it have to be a cuboid and nothing else?” (A cuboid has six faces, all rectangles. Some, but not all of these rectangles may be squares. If all of the faces are squares, it is a cube.)

4. Continue similarly with the other solids. For your convenience, I list their defin-ing characteristics below. All of the descriptions are (I hope) sufficient, but not every detail is necessary for a complete definition.

5. Prism. It has two parallel and congruent faces (the bases), which are both poly-gons. The lateral faces are usually rectangles (making a ‘right prism’), but your set may include a ‘skew prism’ which also has parallel congruent polygons for bases, but the lateral faces are parallelograms. Your set is likely to include a ‘triangular prism’ and a ‘pentagonal prism’ or a ‘hexagonal prism.’ A ‘square prism’ would be called a cuboid.

6. Pyramid. Its base is a polygon, usually a square. Its sides are all triangles, shar-ing a single point as vertex at the top of the pyramid. A ‘triangular pyramid’ is also called a tetrahedron (see below).

Discussion of concepts

297

7. Cone. Its base is usually a circle, and it has just one other surface, a curved sur-face with the base as one edge and a point as the other. There has to be a straight line from the summit to a point on the base. (For your own information, the concept has been generalized to include any closed curve as base. The cone is then ‘swept out’ by a moving straight line from the apex to a point which travels all around the base.)

8. Tetrahedron. This has four faces, all triangles.

Note All of the solids in the sets usually sold are likely to be of regular shapes. I do not think this matters, but I think we should tell the children that these are not the only kind. In topics 12 to 15 they have learned that polygons can be regular or non-reg-ular, so we need just to mention that this is also the case with geometric solids, even though the set only includes regular solids.

This is a straightforward activity which requires children to reflect on their global concepts of the various solids, to analyze them, and to relate them to their existing knowledge of two-dimensional geometry.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 1.16 Classification of geometric solids (cont.)

298

Discussion of concept

Space 1.17 INTER-RELATIONS OF PLANE ShAPES

Concept A variety of part/whole relationships between smaller and larger geometric figures.

Abilities (i) Combining smaller figures to make larger figures. (ii) Finding smaller figures within larger figures.

The ability to see how parts fit together to make a whole, and to find shapes within other shapes, is an important one for later geometry. Here we are concerned with being able to think of, e.g., a parallelogram as made up of two congruent triangles. It is a good example of relational thinking in geometry, analogous to the relationships between numbers which are so important in numerical mathematics.

activity 1 Triangles and polygons [Space 1.17/1]

An activity for up to 6 children. Its purpose is to teach relationships, including angu-lar relationships, between triangles and regular polygons.

Materials • Triangles/polygons set.* These are made by cutting out regular polygons, and then cutting these into isosceles triangles, as illustrated in Figure 32. All the angles should be marked, on both sides.

* Provided in the photomasters

What they do Stage (a) 1. Each child takes a triangle of a different shape. 2. They collect all the triangles which are congruent with the one they have. 3. Each makes a regular polygon with their triangles. (A square may be made using

2 or 4 right angled isosceles triangles.) 4. They examine their polygons, and add together all the angles surrounding the

centres.

Figure 32 Triangles/polygons set

45˚45˚ 45˚

45˚

45˚

90˚90˚

60˚ 60˚

60˚60˚60˚

60˚ 60˚

90˚90˚

45˚45˚

45˚

60˚

60˚

60˚

60˚60˚60˚ 60˚

60˚ 60˚

60˚ 60˚

299

Stage (b) 1. The first child shuts his eyes and takes a triangle. 2. He looks at it and predicts what kind of regular polygon can be made using it,

and how many congruent triangles will be needed. 3. The others help him to collect all the triangles congruent with the one he has. 4. He tests his prediction. 5. If successful he explains how he knew. 6. Steps 1 to 5 are repeated by another child.

activity 2 Circles and polygons [Space 1.17/2]

An activity for up to 6 children. Its purpose is to teach relationships between circles and polygons.

Materials For each child: • Safety compasses. • A protractor. • Pencil and paper. • Scissors.

What they do 1. Each child draws a circle on paper and cuts it out. 2. He folds it along a diameter, then again, and opens it flat.

3. He draws a square.

4. Each child cuts out another circle, and repeats step 2, but folds once more. 5. In this way he draws an octagon.

6. They mark the sizes of angles at the centre. These are arrived at by calculation, not measurement.

7. Next, a hexagon. The circle for this is not cut out.

Space 1.17 Inter-relations of plane shapes (cont.)

300

8. A first point is marked anywhere on the circumference. Using the same length of radius as that by which the circle was drawn, they use the compasses to mark second, third . . . points round the circumference, as shown below. The distance from the sixth to the first point should be found to be the same as all the others.

9. The points are joined to form a hexagon, as above. 10. This is a very satisfying method. Ask “Why does it work?” Hint: join all the

points to the centre. Second hint if necessary: look for equilateral triangles. 11. Next, the equilateral triangle. This uses circles in a different way. 12. Start with a circle.

13. Take any point on the circumference as centre for another circle, using the same radius.

14. This gives two equilateral triangles, of which only one is shown here.

15. Ask: “Do we need to draw the whole circles? If not, how little can we do with?” 16. A pentagon cannot be drawn by a method of the kind above. It is necessary to

calculate the central angle, and use a protractor to draw angles of the required size, as below.

second point

third pointfourth point

fifth point

sixth point first point

72°

72°72°72°

72°

Space 1.17 Inter-relations of plane shapes (cont.)

301

activity 3 “I can see . . . ” [Space 1.17/3]

A game for up to 6 players, but is probably best for 2 or 3. Its purpose is to develop the ability to see figures within figures.

Materials • A complex figure, such as Figure 33. This should be covered with transparent film.*

• Non-permanent markers, a different colour for each player. • Scoring rules on a piece of thin card. • A bowl of counters. * Provided in the photomasters. Alternatively, the figure could be duplicated and

expendable. In this case any kind of coloured marker will do.

Rules of play 1. The figure is circulated from player to player. 2. Each in turn ‘collects’ embedded figures, such as a pair of congruent triangles, a

right angle, a pair of congruent angles or lines, a parallelogram. 3. He names these, and claims them by marking them with his colour. This is done

on or within the shape, as is best in the particular case. 4. If the other players agree, he takes a counter from the pool. 5. The same embedded figure may be claimed in various ways, but not twice in the

same way. 6. The winner is the one with most counters, when no one can think of anything

new to claim and gain agreement thereto. 7. One point is scored for each of the following (By agreement, this list may be

extended.): A pair of parallel lines. An oblong. A pair of congruent triangles. A square. A pair of congruent lines. An isosceles triangle. A pair of congruent angles. An equilateral triangle. A parallelogram.

Figure 33 ‘I can see . . .’

Space 1.17 Inter-relations of plane shapes (cont.)

302

activity 4 Triangles and larger shapes [Space 1.17/4]

A game for up to 6 children, but is probably best for 2 initially. Its purpose is to develop awareness of the relationships between triangles and other shapes.

Materials • A set of cutout triangles, as provided for in the photomasters. For the larger groups a double set will be needed. It is useful to make these sets of different colours.

• Scoring rules on a piece of thin card.

Preliminary Allow the children time to experiment with putting the cutout pieces together in vari-ous ways, and finding out what larger shapes they can make.

Rules of play 1. The object is to collect triangles and make larger shapes, e.g., parallelograms. 2. The triangles are spread out in the middle of the table. 3. Each player in turn takes a triangle, with which (from the second round on) he

starts building larger figures. 4. When there are no longer enough triangles for each player to take one more, one

more round is played as in step 5. 5. In this final round, each player may if he wishes exchange one of his triangles

for one in the pool. 6. Players then count up their scores, each in turn explaining to the others what

points he claims and why. 7. The scoring rules are as follows: composite parallelogram 1 composite rhombus 1 composite rectangle 1 composite square 1 composite right angle 1 composite straight line 1 composite kite 1 composite oblong 1

The same figure may be scored in as many ways as possible. Example:

composite oblong 1 composite rectangle 1 composite parallelogram 1 2 composite right angles 2

total 5

composite rhombus 1 composite parallelogram 1

total 2

Space 1.17 Inter-relations of plane shapes (cont.)

303

composite rhombus 1 2 composite right angles 2 composite parallelogram 1 composite rectangle 1 composite square 1

total (if he says them all) 6

3 composite parallelograms 3 2 composite straight lines 2

total 5

Players may rearrange their pieces at any time up to their final declaration.

In Activity 1, physically putting shapes together is used as the means for learning part-whole relationships. Activity 2 takes a step towards pencil-and-paper geome-try. It is still based on physical manipulation, but calculation and inference are also used. Activity 3 diversifies from regular polygons to all kinds of larger shapes. Prediction is also involved: triangles are chosen because, by combining them with other shapes in imagination, the player expects a desired result. He has to commit himself to taking the piece before this can be tested. So in these three activities, we have another clear example of Mode 1 schema building and testing. The preliminary to Activity 4 consists of combining the cutout triangles physi-cally, and observing the outcome. The activity itself involves combining the shapes first in imagination, and then testing the prediction physically. So here we have a very clear example of Mode 1 schema building followed by testing.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 1.17 Inter-relations of plane shapes (cont.)

Discussion of activities

304

Space 1.18 TESSELLATIONS

Concept That of a tessellation.

Ability To discover whether a given shape will tessellate, or not.

A tessera is one of the pieces of marble, glass, or tile from which a mosaic pave-ment is made, and tessellating is combining these so as to form a mosaic. The everyday meaning of tessellation (which is centuries old) and the mathematical meaning are the same; except that mathematicians, as one might expect, do it on paper. So the transition from cardboard or plastic tessella (little tessera) to draw-ing, which we follow in this topic, also follows its historical sequence. A shape can be tessellated if a number of them, all the same size, can be fitted together (rotated if necessary) to cover a surface without gaps. Here we shall con-cern ourselves with tessellations using one shape only.

activity 1 Tessellating regular polygons [Space 1.18/1]

A group investigation for up to 6 children, working singly or in pairs. Its purpose is to teach the concept of tessellation, and introduce some of the easier examples.

Materials • About 20 of each of the following regular polygons,* cut out of cardboard or plastic, and stored in separate plastic bags. Some of those in each bag have their interior angles marked, on both sides. If the polygons in a bag can be of assorted colours, so much the better.

• Equilateral triangles. • Squares. • Pentagons. • Hexagons. • Octagons. * Provided in the photomasters

What they do 1. Using the equilateral triangles, demonstrate what is meant by tessellation. 2. Each child or pair then takes a bag. 3. Their first task is to discover, by experiment, whether the polygons they have

can be tessellated, or not. 4. Their second task is to find out why they can or cannot be tessellated. The

marked angles provide a clue. 5. In turn, the children tell the rest of the group what they have found out.

Discussion of concept

305

activity 2 Tessellating other shapes [Space 1.18/2]

A group investigation for up to 6 children, working singly or in pairs. Its purpose is to expand children’s concept of tessellation to include rectangles, parallelograms, all triangles, and a variety of other shapes. It also takes children from physical tessella-tion to doing it on paper.

Materials For each child: • Plain paper. • Pencil and eraser. • Ruler. • Instruction card for the group, as below and in photomasters.

TESSELLATIONS

Which of these shapes can be tessellated? Which cannot? Rectangles, any shape. Parallelograms, any shape. Triangles, all varieties. What else can you find out? Share out the task among yourselves, and produce a combined report.

What they do 1. They follow the instructions on the card. 2. In fact, all these shapes can be tessellated. 3. Rectangles are fairly obvious. Their right angles fit together to form straight lines. 4. Parallelograms also tessellate quite easily. Opposite sides, being parallel, have

the same direction; so they fit if the parallelogram is slid along any side without rotation.

From this we can see that the angles at each end of a side combine to give a straight line.

5. Any kind of triangle can be tessellated, since two congruent triangles make a parallelogram. To get the second in position, rotate the first half a turn about the mid point of any side.

From this we can see that the 3 angles of a triangle together make a straight line. The dotted larger triangle is similar to each of the smaller ones, since the an-

gles of the original triangle are each reproduced once.

Space 1.18 Tessellations (cont.)

x o o

x o

x o

x o

x

o

xo o

ox x

xz

z z

z

306

activity 3 Inventing tessellations [Space 1.18/3]

An investigation which may be followed singly or in pairs. Its purpose is to open up the concept of tessellation and teach children two ways of generating their own.

Materials For each child: • Plain and squared paper. • Tracing paper. • Pencil and eraser. • Ruler.

What they do 1. Demonstrate one of the methods for generating tessellations described below. 2. Then allow the children time for inventing their own. 3. Let them show and discuss their results. 4. Repeat steps 1 to 3 with the other method. 5. These tessellations may be coloured.

Method 1 Dissection of a rectangle If a rectangle is dissected into two halves of the same shape, these will tessellate.

Squared paper is useful for this. A suitable dissection line may be obtained by work-ing inwards from the two ends, as below.

Method 2 Parasquigglegrams I discovered this method for myself, and have not yet seen it elsewhere, though it

is too simple not to have been discovered by others. The name is my own. Tracing paper is useful for reproducing the squiggles.

This is a nice use for translations.

Start with 2 intersecting squiggles.

Move A without rotation to the other end of one squiggle, taking the other squiggle with it.

Space 1.18 Tessellations (cont.)

a

a a1

307

Now the other way.

Any shape made in this way can be tessellated.

activity 4 Tessellating any quadrilateral [Space 1.18/4]

A more difficult investigation for children working singly or in pairs. Its purpose is to challenge the abilities of the brighter children.

Materials For each child: • Pencil and plain paper. • Tracing paper. • Ruler.

What they do 1. Tell them that it is possible to tessellate a quadrilateral of any shape. (Note: we mean quadrilaterals which are convex not concave nor crossed.)

Since the children have only encountered the first kind so far, we do not need to mention the other possibilities unless they do.)

2. They may like to try to find out how without help. 3. The method is to rotate the quadrilateral about the midpoint of each side in turn.

By drawing a diagonal in a dotted line, and including this in the rotated versions, two sets of tessellating parallelograms result. For clarity this has been omitted from Figure 34.

4. From this we see that the interior angles of any quadrilateral add up to 360 degrees.

5. Let the children draw their own tessellations of a quadrilateral of their own choice.

Space 1.18 Tessellations (cont.)

a a1

a2

308

In a now-familiar sequence, we start with schema building by Mode 1, physical experience, and progress to Mode 3, creativity. Discussion plays an important part as always. This topic is open ended towards creativity of an artistic kind. If possible, do show the children some of the pictorial tessellations of the artist M. C. Escher. These are published both in book form, and as reproductions sold in art stores.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 1.18 Tessellations (cont.)

Discussion of activities

Figure 34 Tessellation of an irregular quadrilateral

309

Suggested outline for the introductory demonstration

Space 1.19 DRAwINg NETS OF gEOmETRIC SOLIDS

Concept That of a net of a given geometric solid.

Abilities (i) To draw the net of a given geometrical solid. (ii) To recognize what solid a given net belongs to.

This has been well prepared in the earlier topics, in Volume 1, “My pyramid has one square face . . . ” [Space 1.7/5] and ‘Does its face fit?’ [Space 1.7/6]. It con-tinues the development of the relationships between two- and three-dimensional objects, and Activity 2 calls visual imagination into play.

activity 1 Drawing nets of geometric solids [Space 1.19/1]

An activity for children to do individually, after an introductory demonstration. Its purpose is to introduce children to above concept, and to develop the first ability.

Materials Shared among a group: • A standard set of geometric solids, as already used in topic 16. For each child: • Pencil, paper, eraser.

1. A cuboid is a good example to begin with. Put it on the paper with the largest face down, and draw around this. (In all these diagrams, the shaded area represents the face which is in contact with the paper.)

2. Roll it to the left, over the left-hand edge, and draw around it again.

3. Now roll it back to the centre position, and again over the right-hand edge to the position shown. Draw around this.

Discussion of concept

310

Space 1.19 Drawing nets of geometric solids (cont.)

4 Likewise, again.

5. Back to position 1, roll this time over the upper edge, and draw around this face.

6. Repeat step 5, but this time roll over the lower edge.

7. Remove the cuboid, and we are left with its net:

8. This can be done in several slightly different ways. Here is one of them.

What they do 1. Each child chooses a solid, and draws its net in the way they have just seen. 2. He then writes its name of the solid at the bottom of the paper. 3. Steps 1 and 2 are repeated until nets have been drawn of all the solids. 4. The solids are returned to a central pool, and the papers collected in readiness

for Activity 2.

311

Some other nets:

Net of a cylinder Net of a cone

A sphere does not have a net.

activity 2 Recognizing solids from their nets [Space 1.19/2]

A sequel to Activity 2, also for a small group. Its purposes are to check the nets al-ready drawn, and strengthen the connections between individual solids and their nets.

Materials • The solids and their nets, from Activity 1.

What they do 1. The titles at the bottom of the drawings are folded underneath so that they can-not be seen. The papers are then shuffled and put in a pile centrally.

2. The solids are also put in a central pool. The player whose turn it is to start takes the top paper from the pile, and then takes the solid of which he thinks it is the net.

3. The others say whether or not they agree. 4. This continues until all the papers have been used.

Space 1.19 Drawing nets of geometric solids (cont.)

312

activity 3 Geometric nets in the supermarket and elsewhere [Space 1.19/3]

An activity for a small group. Its purpose is to relate the concept of a net to one of its applications.

Materials • An assortment of empty cereal boxes, and the like. If possible, these should be in matching pairs.

1. The boxes are carefully pulled apart where they have been stuck, so that they can be flattened.

2. Each is then put beside a similar box in its original form, for comparison. These can be seen to be similar to the nets of cuboids, with the addition of tabs by which to stick them together.

3. Some of these pairs could be put on display, if desired .

These are both straightforward activities, providing a simple introduction to the concept of the net of a solid.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 1.19 Drawing nets of geometric solids (cont.)

313

Discussion of concept

MIRRORLINE

•P

•P’

[Space 2] Symmetry and motion geometry

Space 2.1 REFLECTIONS OF TWO-DIMENSIONAL FIGURES

Concept Themirrorimageofapictureorothertwo-dimensionalfigure,alsocalleditsreflection.

Abilities (i) Torecognizewhenonefigureisthereflectionofanother. (ii) Toconstructthegeometricalreflectionofasimplefigure,foragivenmirrorline.

‘Reflection’hasthesamemathematicalmeaningasthatineverydaylife.Theonlydifferenceisthatinmathematicsweusuallyworkintwodimensions,sowhiletheobjectsarevisibleinfullintheplaneofthepaper,themirrorusuallyonlyshowsupasaline.However,thereisnothingtopreventourusingarealmirror,heldalongthemirrorlineperpendiculartothepaper,andthisisagoodwaytobegin.

Mathematically,ifPisanypoint,thenitsmirrorimageP’isontheothersideofthe mirror line and the same perpendicular distanceawayfromit.Thismeansthatthemirror line is the perpendicular bisector of PP’.

Themirrorimageofanydrawingormathematicalfigureismadeupofallthepointsarrivedatinthisway.

Youcancheckthatthisdefinitiondoesinfactgiveamirrorimagewhichcoincideswiththatobtainedbyholdingamirroredgeontothepaper,alongthemirrorline.Italsogivesthesameresultastheopticalone,basedonhowlightisreflectedfromamirrorandhowoureyesandbraininterprettheresult. Theterms‘mirrorimage’and‘reflection’(noun)areinterchangeable.Themeaningof‘flip’isslightlydifferent,andisdiscussedintopic10.UntilthenIsuggestthatitisbetternotused.

314

Activity 1 Animals through the looking glass [Space 2.1/1]

Anactivityforasmallgroupofchildren.Itspurposeistointroducetheconceptofamirrorimageinaneasilyrecognizableembodiment.

Materials • Apackofanimalcardsforthegroup.* • Asmallmirrorforeachchild.† *Providedinthephotomasters.Thisconsistsofthreepairsofpicturesofeachof

fivedifferentanimals,makingthirtycardsinall.Ineachpair,onepictureisthemirrorimageoftheother.Iftherearemorethanfivechildren,adoublepackmaybeused.Inthiscaseitishelpfultomakethemindifferentcolours,sothattheycaneasilybeseparatedagainforActivity2.

†E.g.,apocket-orpurse-sizedmirror...oraMIRA.

What they do 1. Thecardsareshuffledandspreadoutfaceupwardsonthetable. 2. Eachchildthenchoosesadifferentanimal,andcollectsallthecardsforthat

animal. 3. Theythentakeanyoneoftheircards,andfindanotherwhichlookslikethe

reflectionofthefirstoneintheirmirror.(Thecardsshouldbeflatonthetable,anedgeofthemirroronthetable,andtheplaneofthemirroratrightanglestoit.Thisisimportantforwhatwillfollowinlateractivities.)

4. Theycontinueuntiltheyhavepairedalltheircardsinthisway. 5. Furtherexperimentshouldbeencouraged,withsharingofwhattheyhavefound

outandverificationbytheothers.E.g.,itispossibletopositionthecardswithoneoneachsideofthemirror,insuchawaythatthesecondcardiswherethemirrorimageappearstobe.Whenthemirrorisremoved,thesecondcardtakestheplaceoftheimage.Whathappensifthemirrorisheldfacingtheotherwayround?Andwhatifoneoftheanimalsisnotpointingdirectlytowardsthemirror?

Activity 2 Animals two by two [Space 2.1/2]

ThisgamefollowsdirectlyonfromActivity1,anditspurposeistoconsolidatethenewlyformedconceptofamirrorimage.Thethirtycardpackissuitableforagroupofuptofourchildren,abovewhichitisbesttohaveasecondpackandanothergroup.

Materials • Animalspack,asforActivity1. • Asinglemirrorshouldnowsufficeforthewholegroup.

1. Theobjectofthegameistocollectpairsinwhicheachcardisthemirrorimageoftheother.Thisisdoneasfollows.

2. Thecardsareshuffled,andputinapilefacedownonthetable.Thetopcardisturnedfaceupandputseparately.

Space 2.1 Reflections of two-dimensional figures (cont.)

Rules of the game

315

3. Inturn,eachplayerturnsoverthetopcardoftheface-downpile,andputsitfaceupseparatelyfromtheotherface-upcard(s).Ifheseesapair,hetakesitandputsthecardssidebysidewheretheotherscansee.Iftheothersagree,hekeepsthepair.Itisalwaysgoodtocheckbyusingthemirror.

4. Morepairswillgraduallyshowasmorecardsareturnedface-up. 5. Theplayerwhohasjustturnedacardhasfirstchancetomakeapair.However,

ifapairisoverlooked,theplayerwhoseturnitisnextmayclaimitbeforeturningoveranothercard.Ifhisclaimiscorrect(asagreedbytheotherplayers),hemayalsohavehisturnasinstep3.Ifhoweverheisfoundtohaveclaimedincorrectly,heloseshisturn.(Thisistodiscouragerecklessclaiming.)

6. Ifatanytimethereisnoface-upcardleft,theplayerwhoseturnitismayturnovertwocards.

7. Whenallthecardshavebeenturnedover,thewinneristheplayerwhohasthemostpairs.

Activity 3 Drawing mirror images [Space 2.1/3]

Anactivityforchildrentodoindividually,checkingtheirresultswithothersintheirgroup.Itspurposeistointroducethemtothemethodforconstructingmirrorimagesaccordingtothemathematicaldefinitiondescribedinthediscussionofconcepts,above.

Materials • Aworksheetforeachchild,seeFigure35.* • Severalmirrorsmaybesharedbyasmallgroup:itisnotessentialforthemto

haveoneeach. • Colouredfelt-tippens(fine),whichmaybesharedbyasmallgroup. *PhotomasterprovidedinSAIL Volume 2a.

What they do 1. Fullinstructionsaregivenontheworksheet,asillustratedinFigure35.

Activity1introducesthemathematicalconceptofreflectionatanintuitivelevel,closelyrelatedtotheireverydayexperience.Strongcuesareprovidedinthepictures,sothechildrenprovideforthemselvesasuccessionofexamplesofmirrorimages,whichtheycheckexperimentally(Mode1)andbydiscussionwithpeers(Mode2).Activity2isagameforconsolidatingtheconcept,usingthesamematerials.InActivity3,theuseofsquaredpaperhelpsthemtomaketheconceptexplicit,andtodiscoverandusethemathematicalmethodforconstructingthem.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 2.1 Reflections of two-dimensional figures (cont.)

316

Space 2.1 Reflections of two-dimensional figures (cont.)

Figure 35 ‘Drawing mirror images’ worksheet

MIRROR OBJECT LINE IMAGE

On the left is an arrow, and on the right is its mirror image. Check this by holding a mirror on the mirror line. Then use the dots to work out how the image was drawn. Now see if you can draw the mirror image of the check mark. Start with the dots. Check two ways: with your mirror, and by comparing your drawing with someone else’s.

Do the same with this one.(Be careful!)

From here on, provide your own dots. You may be able to manage without them.

Continue with the rest on this page, checking each of your drawings in the two ways de-scribed above.

When you have finished, you may like to colour over the lines with a felt tip pen (not too thick). Use the same colour for each object and its image, and a different colour for each pair.

317

Discussion of concepts

Space 2.2 REFLECTION AND LINE SyMMETRy

Concept Linesymmetry.

Abilities (i) Torecognizewhetherornotafigurehaslinesymmetry. (ii) Toindicateallitsaxesofsymmetry.

Afigureissymmetricalifonehalfofitisthemirrorimageoftheotherhalf.Inthiscase,themirrorlineiscalledtheaxisofsymmetry.

Somefigureshavemorethanoneaxisofsymmetry.Wesaythatitissymmetricalabouteachoftheseaxes.Sointhisillustration,theletterAhasoneaxisofsymmetry,theletterXhastwo,andthepentagonhasfiveaxesofsymmetry.

Thesefiguresarenotsymmetrical.

Ifafigureissymmetrical,thewholeofitlookslikeitsreflectionwhereverthemirrorisplaced.Thisishowevernotthedefiningproperty,butaconsequencethereof.Itdoesnotshowthepositionoftheaxisofsymmetry,norhowmanyofthesethereare.ForthisreasonIsuggestthatwedonotmentionittothechildrenunlesstheynoticeitthemselves.Iftheydo,wemighttellthemthatthisisanotherinterestingpropertyofsymmetry—asindeeditis. Inthistopic,‘symmetry’meanslinesymmetrysincethisistheonlykindwhichthechildrenhavemetsofar.Intopic9theywilllearnaboutrotationalsymmetry,andfromthenonweshallneedtosaywhichkindwemean.

Activity 1 introduction to symmetry [Space 2.2/1]

Ateacher-leddiscussionforasmallgroup,afterwhichthereareexamplesheetsforthemtodoontheirown.Itspurposeistointroducethemtotheconceptsdescribedabove.

Materials Foreachchild: • AsmallmirrororaMIRA,asusedinSpace2.1/1. • Anexamplessheet.* • Pencilandpaper. Forthegroup: • Asetofsixcardsshowingthreesymmetricalfiguresandthreenon-symmetrical

figures.* *Providedinthephotomasters.Thisincludesfigureswithone,two,andthreeaxes

ofsymmetry,andalsosomenon-symmetricalfigures.

A

X

R

318

Stage (a) 1. Spreadoutthesetofsixcardsfaceup,andtellthemthattherearetwodifferent

kindsoffigurehere,threeofeachkind.Cantheyseewhichtheyare? 2. Acceptanddiscussanyreasonableanswers.Wehopethatatleastoneofthem

willperceivethesymmetrical/non-symmetricaldifference,inwhichcaseaskhertogrouptogethertheoneswhicharealike.Thenaskiftheycandescribethedifference.

3. Ifnoonegivesthedesiredresponse,groupthesymmetricalfigurestogether,andlikewisethenon-symmetricalfigures,andaskiftheycanseethedifference.Iftheycannot,theyareprobablynotreadyforthisconcept.

4. Useamirrortoshowthemthatforthreeofthefigures,onehalfisthemirrorimageoftheother,bothwaysround.Forthis,themirrorhastobeintherightposition.

5. Letthemexperimentforthemselves.Iftheydonotfindoutforthemselves,askthemwhetherthereareanyofthefiguresinwhichthemirrorcanbeinmorethanonepositiontogivethisresult.Theyshouldalsotryusingthemirrorwiththenon-symmetricalfigures.

6. Tellthemthatthenameforthepropertytheyhavebeeninvestigatingissymmetry,andthatthemirrorlinesforthesearecalledtheiraxes of symmetry.

Stage (b) 1. Theyeachreceiveanexamplessheet,andfollowtheinstructionsthereon.

Activity 2 Collecting symmetries [Space 2.2/2]

Agameforasmallgroupofchildren.Itspurposeistoconsolidatetheconceptofsymmetry,andtogivepracticeinfindingalltheaxesofsymmetry.

Materials • Apackof48symmetrycards.* • Mirrors.† • Pencilandpaperforeachchild. *Providedinthephotomasters.Thepackconsistsof48cards,ofwhich32are

symmetricaland16arenon-symmetrical. †AsusedinActivity1.

Introduction Isuggestthattheyarefirstgivenapreliminarylookatthecards.Thiscanbedonebyspreadingoutsomeofthem,faceup,andinvitingthemtotakealook.Arethereanycardswithmorethanthreeaxesofsymmetry?(Yes:uptoeight.)Arethereanynon-symmetriccards?(Yes:16outofthe48arenon-symmetrical.)

1. Theobjectistocollectfigurestotallingasmanyaxesofsymmetryaspossible. 2. Thepackisshuffledandturnedfacedown.Thetopthreecardsarethenturned

andlaidfaceup,separatelysothatallfacesareshowing. 3. Thefirstplayerpicksupwhichevercardshelikes,fromamongtheface-up

cards.Atthisstageshehastochoosebylookingatit,withoutusingthemirror.(Thisrulewasintroducedwhenwefoundthatsomeplayerswantedtotestallthreecardscarefullywiththeirmirrors,beforechoosing.Meanwhiletheotherplayersgotboredwithwaiting!)

Suggested outline for the discussion

Space 2.2 Reflection and line symmetry (cont.)

Rules of the game

319

4. Next,sheturnsoverthenextcardfromtheface-downpile,andputsitfaceuptoleavethreecardsforthenextplayer.

5. Shethenwritesdownthenumberofaxesofsymmetryofthecardshehasjusttaken,usingthemirrorifdesired.Meanwhilethenextplayerismakingherchoice.

6. Ifatanytimeallthreeface-upcardshavenon-symmetricalfigures,theplayerwhoseturnitismaydeclare“Nosymmetry.”Sheisthenentitledtoputasideallthreeofthese,andlayourthreenewcardsforherselftochoosefrom.Thegamethencontinuesasbefore.

7. Steps3and5arerepeateduntileachplayerhasfivecards. 8. Theplayersthentotaltheirscores,andcheckthoseoftheirneighbours. 9. Ifthereisdisagreement,allplayerscheck,withultimaterecoursetoyourselfas

referee. 10. Anotherroundmaythenbeplayed.

Theseactivitiesrelatetheintuitivepropertyofsymmetrytothepreviouslyformedconceptofreflection,andtherebybringitintoamathematicalcontext.Activity1isforbuildinguptheconceptfromavarietyofexamples.Activity2consolidatesit,withfurtheremphasisonmultipleaxesofsymmetry.Italsodiminishesdependenceonthemirror,sincetheinitialchoicehastobemadesimplybyexaminingthefigure.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 2.2 Reflection and line symmetry (cont.)

Collecting symmetries [Space 2.2/2]

320

Space 2.3 TWO KINDS OF MOVEMENT: TRANSLATION AND ROTATION

Concepts (i) Translation(insomecontextsthesemaybecalledslides). (ii) Rotations(turns).

Abilities (i) Todistinguishbetweenthesetwokindsofmovement. (ii) Tousetherightonefortherequiredpurpose.

Childrenwillalreadyhavetheseconceptsataneverydaylevel.Thepresenttopicintroducestheminamathematicalcontext,andemphasizesthedistinctionbetweenthem.Also,ineverydaylifeleftandrightturnsarenormallyrelativetooneself.Herethechildrenarelearningtoapplythemtooutsideobjects,namelytheirmarkers.

Activity 1 Walking to school [Space 2.3/1]

Agamefortwotofourchildren.Itspurposeistoemphasizethedistinctionbetweentheabovekindsofmovement.

Materials • Gameboard,seeFigure36.* • Fourdifferentlycolouredcounters,witharrows. • Die,onwhichtwosidesaremarkedF(goforward),twosidesR(turnright),and

twosidesL(turnleft). *Providedinthephotomasters.Thefull-sizegameboardhasbeenmadefairlylarge

becauseoftenmorethanonewillbewaitingatthesamejunction.

Discussion of concepts

Figure 36 ‘Walking to school’ map

321

Space 2.3 Two kinds of movement: translation and rotation (cont.)

What they do 1. Theplayersagreeontheirchoicesofhomes,whichshouldbeconvenienttowheretheyaresitting.

2. Eachalsotakesamarker.Thearrowsontheseshowthedirectionsinwhichtheyarefacing.Theyallstartintheirhomesfacingtowardstheroad.

3. Eachinturnthrowsthedie,andmaythendecidewhetherornottomakethemovementindicated.Fallowsthemtogoforwardtothenextjunction(notbeyond),wheretheymustwaituntilitistheirturntothrowagain.RandLrespectivelyallowthemtomakeaturnrightorleft,afterwhichtheymustwaitatthejunctionfortheirnextthrow.(SotheyallneedtothrowFtostartwith,whichgetsthemfromtheirhomesontotheroad.)

4. Theymaygotoschoolbyanyroutetheywish. 5. Thewinneristheplayerwhogetstoschoolfirst.Clearly,thegameshould

continueuntilallhavearrivedatschool!

Notes (i) The board is designed so that there are the same number of junctions between eachhomeandtheschool,bytheshortestroute.

(ii)Theturnsarenotallrightangles,sincewedonotwanttolimittheirconceptofturnstothese.

(iii)Theunstatedassumptionisthatallturnsarelessthan180degrees.However,ifsomeonemakesathree-quarterturntothelefttogivetheequivalentofarightturn,theyarequitecorrectandmaybecongratulatedontheiringenuity.Theplayersshouldthenagreewhetherthisisallowedwithinthepresentgame.

Thisisastraightforwardactivityinwhichthechildrenmanipulatephysicalobjects(markerswitharrows)inwayswhichdirectlycorrespondtomovementsoftheirownbodies.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activity

Walking to school [Space 2.3/1]

322

SPACE 2.4 LINES, RAyS, LINE SEGMENTS

Concepts (i) Thedistinctionbetweenlines,rays,andlinesegments,asthesetermsareusedbymathematicians.

(ii) Parallellines.

Ability Tousetheabovetermscorrectly.

Ineverydaylifeweusetheword‘line’withagreatvarietyofmeanings,suchasrailwayline,telephoneline,airline,andalsoforthekindoflinewedrawonpaper.Thislastisarepresentationofamathematicalline,whichhasnowidthandnothickness,onlylength.Mathematiciansalsohaveavailablethreedifferentnamesforthis,accordingtowhetheritisoffinitelength,orextendsindefinitelyinbothdirections,orextendsindefinitelyinonedirectiononly.Usuallyitisclearfromthecontextwhichwemean,butsometimeswemayneedtobemoreexact,andthepurposeofthistopicistomakeavailableanappropriatevocabulary.

Activity 1 talk like a mathematician (lines, rays, segments) [Space 2.4/1]

Ateacher-leddiscussionforagroupofanysize.Itspurposeistointroducechildrentotheaboveconcepts.

Materials • Pencilandpaper. • Ruler.

1. Withtheruler,drawashortlineandask“Whatdowecallthis?”(“Aline”isthelikeliestanswer.Orbetter,“Astraightline.”)“Forthisactivitywearetalkingonlyaboutstraightlines,notcurvedones.”

2. “Thatiswhatmostpeoplewouldcallit:astraightline. However,Mathematiciansusethreedifferentnamesforthis,sothattheycanbe

quiteclearexactlywhattheymean.Wedon’talwaysneedtousethesenames,butitisusefultobeabletotalklikeMathematicians(withacapitalM)some-times,ifwewant.”

3. “Byaline,Mathematiciansmeansomething which goes on and on withoutend,inbothdirections.Wecan’tdrawthis,soweputarrowsateachendinstead.

4. “Ifwemeansomethingwhichhasabeginningandanend,weputinendpointslikethese.Thisiscalledalinesegment,whichmeanspartofaline.

Discussion of concepts

Suggested outline for the discussion

323

5. “Andthereissomethinginbe-tween,whichhasastartingpoint,butgoesonandoninonedirection.Thisiscalledaray.

6. Whatwouldyoucallthethreesidesofthistriangle?(Linesegments.Theyendateachvertex(corner).Inthiscasewedon’tneedtoputinendpoints—theverticesshowwherethesidesend.)“Canyouthinkofsomeotherexamples?”(Thesidesofasquare,oblong,ofanykindofpolygon.)

7. Whatwouldyoucallthearmsofthisangle?(Rays.)“Why?”(Be-cause the size of an angle is the samewhateverthelengthofthearms.)Sowe’llputarrowsontheendsofthearms.(Thesearenotin-cludedinthepresentfigure,whichshowsthefirststageonly.)

Whentheylearnaboutcompassbearings,theywillrecognizetheseasanotherexampleofrays.Acompassbearingisadirectionfromagivenpoint,thepointofobservation.Thismayrelatetosomethingashortdistanceaway,ortoadis-tantstar.

8. Simpleexamplesoflines,inthemathematicalsense,arehardertothinkof.Thenumberlineisagoodexample,providedthatweincludenegativenumbers,butitistooadvancedforchildrenatthisstage.Suggestionswillbewelcome.HereisonewhichIhadfromLauraBlick,aged10:“Thehorizon,atsea.”Yourownchildrenmightliketodiscussthisone.

Activity 2 true or false? [Space 2.4/2]

Thisisamoredifficultactivity,foragrouppreferablyofthreeorfourplayers.Itisnotessentialatthisstage,butwillexercisethewitsofyourmoreablechildren.Youmightalsochoosetoreturntoitlaterwithotherchildren.Itsmoreobviouspurposeistoconsolidatethedistinctionbetweenlines,rays,andsegments,andtolinkthesewithanacceptednotation.Sinceatthisstageinchildren’smathematicsthisisnotacriticaldistinction,Iwouldseeasamorevaluablepurpose,thoughlessobvious,thatofdevelopingchildren’sabilitytothinkandspeaklogicallyinamathematicalcontext.

Materials • Apackoftwelvecards,sixbearingtheword“True”andsixbearingtheword“False.”*

• Pencil,paper,andrulerforeachchild. *Providedinthephotomasters

Space 2.4 Lines, rays, line segments (cont.)

324

Introductory 1. ThelineABmeansthis: explanation ThelineBAmeansthesame. Notethatalinegoesinboth

directions. 2. TherayABmeansthis:

ButtherayBAmeansthis.

Notethataraygoesinonedirec-tiononly.

3. ThelinesegmentABmeansthis: ThelinesegmentBAmeansthe

same. Notethatalinesegmentdoesnot

normallyhaveadirection.(Ifitdoeswecallitsomethingdifferent.)

The activity 1. Theybeginbyallmakingtheirowncopiesofthisdiagram.Pointoutthatwemayuseanyletterswelikeforlabellingpoints.

(IstoppedusingABCDbecauseIfoundthatwhenspoken,BandDaresome-timesconfused.)

2. “I’mgoingtomakesomestatements,andI’dlikeyoutotellmewhethertheyaretrueorfalse.ThelinePQandthelineRSareboththesameline.Isthistrueorfalse?”(True.)

3. “Hereisanotherstatement.TherayPQandtherayRSareboththesameray.Trueorfalse?”(False.)

4. “Isthefollowingstatementtrueorfalse?ThepointRisonthelinesegmentPQ.”(False.)

5. “Isthefollowingstatementtrueorfalse?TherayPQgoesthroughthepointR”(True.)

6. Ifnecessary,givesomemoreexamplesuntiltheyhavegraspedtheidea.Herearethreemoretruestatements:

ThepointPisontheraySR. ThelinesegmentPSispartofthelineQR. TherayPRgoesthroughthepointS. Andthreefalsestatements. ThepointRisontherayQP. ThelinesegmentPSispartofthelinesegmentQR. TherayQRgoesthroughthepointP. 7. Thecardsarenowshuffled,andeachplayertakesacard.Theylookattheirown

cards,butdonotlettheotherssee. 8. Eachplayerthenwritesdownastatementwhichiseithertrueorfalseaccording

towhatisonhercard.

A B . . A B . .

Space 2.4 Lines, rays, line segments (cont.)

A B . .

A B . .

P Q R S . . . .

325

9. Whenallhavedonethis,theplayerwhoseturnitistostartreadsoutherstate-ment.Theothersinturnsaywhethertheythinkthisistrueorfalse,withdiscus-sionifthereisdisagreement.

10. Steps7,8,and9maynowberepeated. 11. Notethat(e.g.)“RisonPQ”isnotaproperstatement,sincewedonotknow

whetheritistrueorfalseuntilweknowwhetherPQisaline,ray,orlineseg-ment.”“RisonthelinesegmentPQ”wouldhoweverbeacceptableasastate-ment(itisfalse)sinceinthiscontextRcannotbeanythingelsebutapoint.“ThepointR”is,however,preferable(inthepresentactivity).

12. Theymayliketoknowthesequickerwaysofwritingdowntheirstatements.

Activity1introducesthenewconceptsinastraightforwardway,andrelatesthemtogeometricalknowledgewhichtheyalreadyhave.Thisisprobablyasmuchasisnecessaryformostchildren.IwroteActivity2somewhatasanexperiment,andwaspleasantlysurprisedtofindhowwellitwasreceivedbyagroupofbright5thgradechildren.Theylikeditasmuchastheeasieractivitiestheyhaddonewithmeinaprevioussession,saying“It’sachallenge.”

OBSERVE AND LISTEN REFLECT DISCUSS

A B . . A B . .

Discussion of activities

The line AB The ray AB The line segment AB

AB AB AB

Space 2.4 Lines, rays, line segments (cont.)

True or false? [Space 2.4/2]

A B . .

A B . .

P Q R S . . . .

326

SPACE 2.5 TRANSLATIONS OF TWO-DIMENSIONAL FIGURES (SLIDES WITHOUT ROTATION)

Concepts (i)Dimensions:one,two,andthreedimensionalobjects. (ii)Translationsoftwo-dimensionalfigures(slideswithoutrotation). (iii)‘Motionlines’(myterm)whichgivethelocationofeachpointoftheimage

figure.(Thesearewhatinthefuturesomeofthemwillknowasvectors.)

Abilities (i)Tousemotionlinestodrawtheresultofaslideforsimplefigures. (ii)Torecognizethisrelationshipbetweentwofigures.

Intopic1,welearnedabout(mathematical)reflection.Thisisjustoneofseveralmathematicaloperationswhichcanbedoneonageometricfigure.Inthepresenttopicweintroduceanotherone,namely,translation,whichisshortforaslidewithoutrotation;andthiswillbefollowedintopic9byrotation,whichisaturningwithoutaslide.Thesearegeometricalcounterpartsofarithmeticaloperationssuchasadditionandmultiplication,whichactonnumbersandresultinanothernumber.Heretheoperationsareongeometricfigures,andtheresultisanothergeometricfigure,orthesameoneinadifferentposition.Butthesamesequence,‘Start,Ac-tion,Result,’canbeseeninallofthese. Itisperhapsapitythatthesameword‘figure’isusedforageometricfigureandforasymbolrepresentinganumber,i.e.,anumeral.However,mydictionarygivesnolessthantwelvedistinctmeaningsfor‘figure,’oneofwhichis‘adiagramorpictorialrepresentation.’Thisisthekindwearetalkingaboutinthepresentnetwork. Ihavealsotakentheopportunitytointroduce,inasimpleway,theconceptofdimension.

Activity 1 Sliding home in Flatland [Space 2.5/1]

Anactivityforasmallgroupofchildren.Itspurposeistohelpchildrentoformtheaboveconceptsfromanumberofsuitableexamples.Theideaispartlyborrowedfrom a mathematical classic called Flatland: a romance of many dimensions, by a Square(author,E.A.Abbott,1884).

Materials • Gameboard,seeFigure37.* • Setofcutoutpolygons.* • Ruler. *Providedinthephotomasters Note Thepolygonsneedtobecutoutofthickcardboard,sothattheydonotgetstuck

undertheruler.Thelattershouldberigidsothatitstaysflatalongitswholelength.

Discussion of concepts

327

Introduction 1. Flatlandisanimaginarycountryinwhichthereareonlytwodimensions,likeasheetofpaperwithnothickness.Itsinhabitantsarepolygons,whichalsohavenothickness.

2. Thecountryisforthemostpartverycold,coveredwithice,andthereforeveryslippery.Thereisacentralareawhichiswarmandnotslippery,inwhichthepolygonscanmeetandmoveatwill.Theirhomesarealsowarm,butthesearetheexactsizeandshapeofthepolygonswholivethere.Nomovementispossibleinthese.

3. Thepolygonsliketomeeteachotherinthemove-at-willarea.Theproblemisafterwardstogettotheirhomessothattheyfitexactly,sinceanypartswhichprojectwillgetfrostbite.Forthistheyhavetopositionthemselvessothatoncetheypushofffromthemove-at-willarea,theycangethomebyaslidewithoutrotation.Theyarenotcleverenoughtodothisforthemselves.

4. Thecountryisgovernedbyaruler,oneofwhosemostimportantfunctionsistokeephissubjectssafebyhelpingthemtotakeuptherightstartingpositionsfromwhichtheycangethomebyaslidewithoutrotation.Sinceitisnotpossi-bletoseethehomesfromthecentralarea,successfulrulersarehardtofind.

Figure 37 Game board for ‘Sliding home in Flatland’ [Space 2.5/1]

Space 2.5 Translations of two-dimensional figures (slides without rotation) (cont.)

328

Rules of the game 1. We,ourselves,arefortunateenoughtohavethreedimensions—wehaveheight,

width,andthickness.SowecanlookatFlatlandfromabove,whichmakesthetaskofpositioningthepolygonsmucheasier.Wealsohavethehelpofanotherkindofruler,thesortusedinmathematics.

2. Playersbeginbytakingpolygonsinturnuntiltherearenotenoughleftforan-otheroneeach.(Theonesweusedohavethickness,orwewouldnotbeabletohandlethem.)

3. Theplayerwhoseturnitishastheruler.Heputsoneofhispolygonsinthemove-at-willarea,andpositionstherulerinthecorrectpositionforhelpingthisparticularpolygontotakeuptherightstartingposition.

4. Withhelpfromtheplayer,thepolygonpositionsitselfagainsttheruler,asinthediagramabove,andthendoesaslidewithoutrotationinthedirectionofitshome.Itisarelieftobothpolygonandrulerifhereachestherighthomeintherightposition,butinanycasethepolygonhastostaythereandtheturnpassestothenextplayer.

5. Thegamefinisheswhenallthepolygonsarehome.Anotherroundmaythenbeplayed.

Note Whenthechildrenarefamiliarwiththegame,theywillbegintonoticethatsomeofthepolygonscanfitintotheirhomesinmorethanoneposition.Intheabovefigure,thepolygonagainsttherulerhastwopossiblepositions,andthesquarewhosehomeisjustabovehasfour.Thisrelatestothenumberofrotationalsymmetries,whichtheywillstudyinmoredetailintopic9.

Follow-up Ifwearethree-dimensional,andFlatlandistwo-dimensional,whatwouldbe discussion one-dimensional?(Aline.Inmathematicsalinehasnowidthandnothickness,

onlylength.)Canyouthinkofsomethingwhichhasnodimension?(Apoint.Thishasonlyposition,likeadotonpaper.Ithasnolength,nowidth,andnothickness.Wearenowtalkingaboutamathematicalpoint,whichexistsonlyinourimagina-tion.Adotonpaperdoeshavesomesizeorwewouldnotbeabletoseeit.)

Someofthechildrenmightenjoyfollowingtheirmathematicalimaginationalittlefurther,asdidtheauthoroftheoriginalbook.

Space 2.5 Translations of two-dimensional figures (slides without rotation) (cont.)

329

Activity 2 Constructing the results of slides [Space 2.5/2]

Anactivityforchildrentodoindependently,usingworksheets.Itspurposesare(i)forthemtoexamineinmoredetailwhathappenstofigureswhichundergotrans-lations (slides without rotations) and the relationship between each point of the originalfigureanditsimage;and(ii)forthemtobeabletoconstructgeometricallytheresultsoftranslations,forsimplefigures.

Materials Foreachchild: • Worksheet.* • Pencil,ruler,eraser. *Twoareprovidedinthephotomasters,oneforStage(a)andoneforStage(b).

Suggested 1. Explainthatsofartheyhavemadeslidesbyactuallyslidingpiecesofcardboard introduction invariousshapes.Nowwemoveontoslideswithoutrotationoffiguresdrawn

orprintedonpaper. 2 AskthemtolookatWorksheet1,thinkabouttheiranswerstothequestion

“Whatcanyousay...?”,butkeepquietuntilallhavehadtimetothink.Youcantheninviteanswers,inturn.

3. Thefirstpointisthatallthe‘lines’areinfactlinesegments.Theyareofadefi-nitelength,withendpoints.

4. Second,theyareallthesamelength. 5. Third,theyareparallel. 6. Whydotheyhavearrows?Becausetheslideisfromonelocationtotheother.

Thereisastartingpositionandafinishingposition,andthisiswhatthearrowsaretheretoshow.ForthisreasonIcallthem‘motionlines.’(Thisfitsinwellwiththetermusedinmoreadvancedmathematics,whichis‘vector.’)

Stage (a) Theyshouldnowbereadytocompletethefirstworksheet,comparingresultsanddiscussingifnecessary.Somechildrenmayinventotherwaysofdrawingtheim-agefigure,whichmaybediscussedalsotoseeiftheygiveacorrectresult.How-ever,theyshouldalsolearntheoneindicatedontheworksheetsincethisfitsmorecloselywiththeconceptsofthepresenttopic.

1. WhyisthereonlyonelinetoshowtheslidesforthecheckmarkandtheletterF?(Thisisallthatisneeded.Theothermotionlinesarejustthesameexceptfortheirstartingpoints.)Whatdowemeanby“justthesame”?(Theyarethesamelengthanddirection.)

2. (Optional)Theymayliketoknowthatdirectedlinesegmentsofthiskindhaveaspecialname.Thesearecalledvectors,andareimportantinadvancedmath-ematics.

Stage (b) ThisusesWorksheet2,andshouldnotbeattempteduntiltheyhavecompletedActiv-ity3.Itismoredifficult,andyoumaywishtopostponeituntillater.

Suggested follow-up discussion

Space 2.5 Translations of two-dimensional figures (slides without rotation) (cont.)

330

Suggested introduction

Activity 3 Parallels by sliding [Space 2.5/3]

Anactivityforchildrentopractiseindividually,afteryouhaveshownthemhow.Itspurposeistoequipthemwithanewskill:drawingparallelstoagivenlineinanydesiredposition.Thisisausefulabilityformanygeometricpurposes,andhasanimmediateuseinStage(b)ofActivity2.

Materials Foreachperson • Ruler. • Set-square. • Pencilandscrappaper.

1. Demonstratethemethodfordrawingoneormorelinesparalleltoagivenline,explainingasyougoalong.

2. Firstwepositiontheset-squarealongthestartingline(shownthicker,atthetopofthediagram).

3. Weholdtheset-squarefirmlyinposi-tion,andbringtheruleragainstit.

4. Fromnowon,itistherulerwhichhastobeheldfirmlyinthisposition.

Weslidetheset-squarealongit(inthesamewayaswedidwiththepoly-gonsin‘Flatland’)toanypositionwechoose.Keepitherewithasparefin-gerorthumb,anddrawalinealongit.Moveitsomemore,anddrawanotherline.Thelinescanbedrawnofanylength,andneednotalwaysstartupagainsttheruler.

What they do 1. Theyshouldpractisetheaboveuntiltheyareproficient. 2. Next,theyshouldexperimentstartingwiththehypotenuseagainstthestarting

line.Sometimesthismaybemoreuseful. 3. Nowtheyshouldsetthemselvesthetaskofdrawingalineparalleltothegiven

linethroughagivenpoint,experimentingwithdifferentpositionsofthispoint. 4. Theyshouldpractisetheforegoinguntiltheyareproficient.Theyarethenready

toreturntoActivity2,Stage(b).

1. Whydoesitwork?(Becausewhenanyobjectslideswithoutrotation,thedi-rectionofallitslinesstaysthesame.Lineswhichhavethesamedirectionareparallel.)

2. Arewetalkingaboutlines,rays,orlinesegments?(Allthree.Ineverydaylan-guage,wedon’thaveawordwhichmeansboth‘he’and‘she.’Itwouldbeuse-fulifwedid.Likewise,whenthemathematiciansattachedaspecialmeaningsto‘line,’‘ray,’and‘linesegment,’theyleftuswithoutawordwhichincludedallthree.Mypersonalviewisthattheyshouldhavethoughtaboutthis!)

Suggested discussion of what they have done

Space 2.5 Translations of two-dimensional figures (slides without rotation) (cont.)

331

Discussion of activities

Space 2.5 Translations of two-dimensional figures (slides without rotation) (cont.)

Parallels by sliding [Space 2.5/3]

Thefirstactivity,‘SlidinghomeinFlatland,’embodiestheconceptsinphysicalactivitieswhicharesimplebutcapturethechildren’sinterest.Thesecondactiv-ityisappreciablymoredifficult.Itmakesexplicitmuchwhichwasimplicitintheactivitywithmanipulatives,andalsogivesanopportunitytoaskthemtoverbalizethese.InStage(a)theemphasisisonthemethod(drawingthemotionlines,whichareparallellinesegmentsofequallength)ratherthangettingtheparallelsexact.Activity3introducesthemethodfordrawingtheseaccurately,whichisitselfanotherapplicationofslideswithoutrotation.TheyaretheninapositiontouseitinStage(b)ofActivity2.Thisisfairlydemanding,butthewayhasbeenwellprepared.

OBSERVE AND LISTEN REFLECT DISCUSS

332

SPACE 2.6 DIRECTIONS IN SPACE: NORTH, SOUTH, EAST, WEST, AND THE HALF POINTS

Concepts (i) North,south,east,west,asdirectionsinspacewhichareindependentofwhereweareontheearth’ssurface.

(ii) Thefourhalfpoints,northeast,etc.

Abilities Touseacompassandcompassdirectionstoidentifyandmakemovementsinspecifieddirections.

Intheprevioustopicweintroducedchildrentothedistinctionbetweentwokindsofmovement,translationandrotation.Theideaofdirectionisanimportantpreparationfordevelopingbothoftheseinfurtherdetail,asweshallseeinlatertopics.Compassdirectionsprovideanexcellentstartingpoint,withimportantandclearreal-lifeapplications.

Activity 1 Compass directions [Space 2.6/1]

Ateacher-leddiscussionforagroupofanysize.Itspurposeistoconsolidate,organize,andexpandchildren’severydayknowledgeofthecompass,andcompassdirections.

Materials • Amagneticcompass.

1. Whatisthis?Whatisitfor? 2. Howdoesithelpustofindourway?Whatelsedoweneed?(Amap.But

we must also be able to match the directions on the map with directions on the ground,andacompassissometimestheonlywayofdoingthis.Wealsoneedtoknowwhereweareonthemap,andacompasscanbeusedforthisalsoifweknowhow.)

3. Whatarenavigators?Wheredotheywork?(Inshipsandinaircraft.Theirjobistotellthepilotinwhichdirectionstosteer,sothattheymayallreachtheirdestinationsafely.)

4. InwhatdirectionisNorth?(...fromwheretheyarenow) You might also wish to tell them about: 5. Magneticnorthandtruenorth. 6. Thepolestarandthenorthpole.(Astarchartisusefulhere.) 7. Theapparentmovementofstars.(Becauseoftherotationoftheearth,which

alsobringsdayandnight,thestarsappeartorotatewhenseenfromtheearth.Justonestarappearsstill,andthisistheonewhichtheearth’saxisofrotationpointstowards.Oneendofthisaxisofrotationisthenorthpole,andtheotheris...?Sothepolestarshowsusthetruenorth.)

Discussion of concepts

Suggested outline for the discussion

333

Activity 2 Directions for words [Space 2.6/2]

Anactivityforasmallgroupofchildren,sayuptosix,ormoreifenoughmaterialsareavailable.Itspurposeistoconsolidatechildren’sknowledgeofthefourmainpointsofthecompass,andthefourhalfpoints.

Materials Foreachplayer: • Agameboard,asillustratedinFigure38.* • A two-sided directions card showing on one side the eight main compass

directionsandontheothershowingonlyNfornorth,alsoillustratedinFigure38.*

Forthegroup: • Twosetsofwordcards,easierandharder.* *Providedinthephotomasters

What they do Stage (a) 1. Eachplayerhasagameboard,andputsherdirectionscardonthemiddlewith

thenorthdirectionpointingtowardsthetoporthecard.Inthiscase,itwillbepointingtotheletterT.Tobeginwith,theyallhavethesideshowingthenamesofallthedirectionsfaceup.

2. Oneplayer,thecaller,alsohasoneofthesetsofwordcards.Shepicksoneoftheseatrandom,whichshelooksatherselfbutdoesnotlettheotherssee.Thecallershouldchoosethewordeitherbycuttingthepackorpickingoneatrandomwiththecardsfacedown,notbylookingthroughandchoosingone.Thelattermakesitveryeasytoguesswhenanotherplayeristhecaller.

3. Shethencallsoutthedirectionsforthelettersinorder.E.g.,“Firstletter,south.”Theotherplayersallwritethis.“Nextletter,northwest.”Theotherplayerswritethis,andsoon,untilthewordiscomplete.

4. Thecallerthenshowsthewordtotheotherplayers.Ifsomedonothavethesameword,sherepeatsthedirectionsgiveninstep3whiletheotherscheckwhattheyhavewrittentofindwhetheritwastheyorthecallerwhomadethemistake.

5. Anotherplayerthentakesovertheroleofcaller,andsteps1to4arerepeated.

Figure 38 ‘Directions for words’ game boards and two-sided directions card

Space 2.6 Directions in space: north, south, east, west, and the half points (cont.)

334

Stage (b) ThisisplayedinthesamewayasStage(a)exceptthattheyturnovertheirdirections

cards,sothatthesideshowingistheoneonwhichonlynorthismarked.Thecaller(only)mayifnecessarytakealookattheothersidetoensurethathercallsarecorrect,butwithoutlettinganyoneelsesee.

Variation Thisgoesbestwhenusingtheharderpack.Ifaplayerthinkssheknowstheword,shemayholdupahandandcalloutthenextdirection(nottheworditself).Thecallerwillsaywhetherornotthisiscorrect.Onlyoneguessperplayerisalloweduntilallhavehadachancetoguess.

Notes (i)Theeasierpackcontainsonlywordsoffourorfiveletters,includingsomeplurals.Theharderpackcontainsonlywordsoffivelettersupwards.

(ii)Youmightwishtolookthroughthewordcardsandremoveanywhichyouthinkthechildrenmightnotknow,e.g.,‘arbitrate’.Alternatively,youmightthinkitusefultoleavethesein,toenlargetheirvocabularies.

(iii)IwouldhavepreferrednottoincludetheletterE,sincethisiscommonlyusedasanabbreviationforeast.However,althoughIusedacomputerprogramforgeneratinganagrams,modifiedtoallowthesamelettertwice,Ifoundthatthere were not enough words of more than three letters to be made from a set of eightwithoutE.Toreducethepossibilityofconfusionfromthis,thefourmaincompassdirectionsarethereforespeltinfullonthedirectionscards.

Activity 3 island cruising [Space 2.6/3]

Agameforasmallgroupofchildren,uptothreeorfour.ItprovidesanauticalembodimentoftheconceptsintroducedinActivity2,andfurtherconsolidatestheabilitylistedabove.

Materials • Gameboard,seeFigure39.* • Packofdirectioncards* • Abowlofcounters. • Foreachplayer,adifferentlycolouredmarkerintheshapeofaboat.† *Providedinthephotomasters †Ingenuityisrequiredhere.Ihaveasetmadeofcutoutsfromcolouredcardboard,

withablunttackpushedthroughinthepositionofamast(tomakeiteasiertopickup);andanothersetmadefromhalf-shellsofcashewnuts.Othersuggestionswillbewelcome.

1. Theshipsallstartintheharbouratthesouthendofthemap. 2. Theobjectofthegameistocollectasmanypassengersaspossiblefromthe

islands.Thisisdoneasfollows. 3. Thedirectionspackisshuffledandputfacedownonthetable.Thetopthree

cardsarenowturnedfaceupandputseparatelybesidethepack. 4. Theplayerwhoseturnitismaytakejustoneoftheface-updirectioncards,

andgotothefirstislandwhichisinthatdirectionfromwheresheis.(Inotherwordsshemaynotcontinuepastit.)Atthatislandshemaypickupjustonepassenger,andmustthenstaythereuntilhernextturn.Pickingupapassengerisrepresentedbytakingacounterfromthebowl.

Space 2.6 Directions in space: north, south, east, west, and the half points (cont.)

335

5. Theunshadedislandsareuninhabited.Sinceboatsmayonlysailinthedirectionsshownonthecards,itmaybehelpfultocallattheseonthewaytootherislands.Theymayalsoreturntotheharbouriftheywish,buttherearenopassengerstobepickedupthere.Thereisnoobjectiontotherebeingmorethanoneboatatthesameisland.

6. Ifaplayertakesacard,shereplacesitbyturningoverthetopcardfromtheface-downpacksothattherearealwaysatleastthreeface-upcardstochoosefrom.Ifshedoesnottakeacardshesays“Pass.”

7. Ifawholeroundgoeswithoutanyplayertakingacardfromthethreeface-upones,theseareputasideandthreenewonesareturnedover.

8. Thegamecontinuesuntiltherearenodirectioncardsleftintheface-downpack,anduntilallhavehadachancetousetheexistingface-upones.

9. Thewinneristhentheboat-ownerwiththemostpassengers.

Figure 39 ‘Island cruising’ game board

Space 2.6 Directions in space: north, south, east, west, and the half points (cont.)

Space2.6/3

336

Thereal-worldimportanceofcompassdirectionsiseasilyunderstoodbychildren,andthedetailedknowledgedevelopedinthisandthenexttopiccanbedirectlyappliedtosituationsinvolvingtheuseofmapstofindone’sway,andgivedirectionstoothers. Activity1isforunderliningthisrelationship,afterwhichActivity2developstheconceptsinasmall-scalesituation.Activity3thenenlargesthefieldofapplicationtoaboardrepresentationofalarge-scalesituation.Whenfieldtestingthisactivity,Ifoundthatitdidnotmatterthatsomeofthechildrenweresittingoneithersideoftheboard:theywerestillabletogetthedirectionsright.Onreflection,Irealizedthatthiswasagoodlearningsituation.Althoughinamap,northisnearlyalwaystowardsthetopofthepage,itisnotalwaysthedirectioninwhichwearelookingatoursurroundingswhenwewanttofindourway.Soitisgoodtobeabletoface(e.g.)east,andknowthatwestitbehindus,northonourleft,andsouthonourright.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Space 2.6 Directions in space: north, south, east, west, and the half points (cont.)

Island cruising [Space 2.6/3]

337

SPACE 2.7 ANGLES AS AMOUNT OF TURN; COMPASS BEARINGS

Concepts (i)Anangleasrepresentingaturn. (ii)Anangleasrepresentingthedifferencebetweentwodirections. (iii)Clockwiseandcounterclockwiseturns.

Abilities (i)Todescribeagivenangleintermsofquarter,half,three-quarterturnsandcompleteturns.

(ii)Todothesameintermsofdegreesformultiplesof30degrees,andforhalfarightangle.

IntheSpace1network,angleswereconsideredfromtheaspectoftheirshape.Herewethinkaboutthemintwootherways. First,asadifferencebetweentwodirections.Thisisthesamebothwaysaround.E.g.,theanglebetweennorthandnortheastis45˚,andsoistheanglebetweennortheastandnorth,inthesamewayasthedifferencebetween7and4isthesameasthedifferencebetween4and7. Second,asaturnfromonedirectiontoanother.Thisisnotthesamebothwaysround.Aturnfromnorthtonortheastisaturnof45˚clockwise,butaturnfromnortheasttonorthisaturnof45˚counterclockwise.Wecouldalsousepositiveandnegativesignsifweagreedwhichdirectionwaspositive,butunfortunatelytherearetwoconventionshere.Compassbearingsasusedinnavigation,etc.,takeclockwiseasthepositivedirections;butmathematiciansusecounterclockwiseaspositive,sincethisfitsinbetterwithcoordinateaxes.Thisseemstomeagoodreasonforstayingwithclockwiseandcounterclockwise. Acompassbearingistheclockwiseangleofturnfromnorthtothegivendirection,sozero,quarter,half,andthree-quarterturnscorrespondtothemajorcompassbearingsnorth,east,south,west.Foraccuratedirection-finding,smallerunitsareneeded.TheancientEgyptiansdividedacompleteturninto360˚,andthisistheunitstillinuse.Sothedirectionnortheastcorrespondstoabearingof45˚,andnorthwesttoabearingof315˚. Atthisstage,weareconcernedwithbuildinguptheconcept,andtheuseofaprotractorformeasuringanglestothenearestdegreewillnotbeintroduced.Multiplesofthirtydegreesgivetwelvemajordirections,includingthefourmainpointsofthecompass,andIhavechosentheseasgivingagoodoverview,andaframeworkwithinwhichmoredetailcaneasilybeintroducedasandwhenrequired.

Discussion of concepts

338

Suggested outline for the discussion

Activity 1 Directions and angles [Space 2.7/1]

Ateacher-leddiscussionforasmallgroup.Itspurposeistointroducetheideaofanangleasrepresentingthedifferencebetweentwodirections,oraturnfromonedirectiontoanother.

Materials • Twopencilsforeachparticipant,includingteacher.(Ballpointswilldoequallywell.) • Scrappaperforeachchild.

1. Askthemaquestionwhichislikelytobeansweredbypointing,suchas“Showmewherethefireextinguisher/clock/mydeskis.”Thisisoneofthemostcommonwaysinwhichweshowadirection.Onpaper,weuseanarrow.Inourpresentdiscussionwearegoingtousepencilsaspointers.

2. Putyourownpencilonthetable,andaskallthechildrentoputtheirpencilssothattheyallpointinthesamedirectionasyours.Repeatafewtimeswiththechildrentakingturnstosetthedirection.Ineachcasethepencilsshouldallbeparallel,withthesharpendspointinginthesamedirection.

3. Nowputtwopencilswiththeunsharpenedendstogetherandthepointedendsapart,toformanangle.“Hereweareshowing....”(twodirections)“Andthedifferencebetweenthesetwodirectionsis?”(anangle)

4. Askthechildrentoshowotherpairsofdirectionswithroughlythesameanglebetweenthemasyours.Repeatafewtimeswithotherchildrensettingtheangle.Emphasizethatwearenottryingtobeexact:theideaissimplytoshowthatdifferentpairsofdirectionscanmakethesameanglebetweenthem.

5. Nowputthepencilstoformarightangle.“Doesanyoneknowwhatthisangleiscalled?”

6. Likewiseforastraightangle. 7. Nowwewanttointroducetheideaofanangleasshowingaturn.Putapencil

toshowastartingdirection,andanothernexttoit.Rotatethesecondpencilthroughafullturn,andsay“Thishasmadeafullturn,allthewayaround,untilitfinishedinthesamedirectionasitstarted.”

8. Nextputtwopencilsatrightanglestoeachother,andaskhowmuchofafullturnthisshows.(aquarterturn)Sothesearetwonamesforthesameangle.

9. Askthesamequestionfortwopencilsendtoendinoppositedirections.(ahalfturn,equaltoastraightangle)

10. Fromthepositioninstep9,rotateoneofthepencilsclockwiseuntilthetwoareagainperpendicular.Thisangleneedsdiscussion,sinceitcanrepresenteitheraquarterorathree-quarterturndependingonthestartingandfinishingpositionsandthedirectionofrotation.(Theseareeasiertodemonstratethantodescribe.)

11. Completeafullturnclockwisewiththesamepencilrotating,andelicittheideathatafullturnisthesameasfourrightangles.

12. Next,wewantthemtoconnecttheforegoingwiththefourmaincompassdirections,whichtheyalreadyknow.E.g.,“Whatistheanglebetweeneastandsouth?Betweensouthandnorth?Betweeneastandnorth?”Thelastdependsonwhethertheturnisclockwise(athree-quarterturn)orcounterclockwise(aquarterturn).Explainthatwhenusingcompassdirections,turnsarealwaysclockwiseunlesswesayotherwise.Sotheanglebetweenwestandsouthisathree-quarterturn,whichisthesameasthreerightangles.

13. Finally,wewanttorelateallthesetothemeasurementofanglesindegrees. 14. Explainthatfordescribinganglesofothersizesthantheabove,weusedegrees.

Wealsousedegreeswhengreateraccuracyisneeded.Arightangleis90degrees,astraightangleisa180degrees;soathree-quarterturnis?Andafullturn,is?

Space 2.7 Angles as amount of turn; compass bearings (cont.)

339

Activity 2 Different name, same angle [Space 2.7/2]

Agameforasmallgroup.ItsobjectisconsolidatetheideasencounteredinActivity1,andespeciallytorelateallthedifferentwaysofnamingthesameangle.Itisplayedinmuchthesamewayas‘Collectingsymmetries,’Space2.2/2.

Materials • Gamepackofcards.* *Providedinthephotomasters.Thisshouldbecopiedtwice,togiveapackof40

cards.

Introduction Isuggestthattheyarefirstgivenapreliminarylookatthecards,byspreadingoutsomeofthem,faceup,andinvitingthemtotakealook.(Thisisagoodtimetoexplainthemeaningofthesignfordegrees.)Arethereanycardswhichhavedifferentnamesforthesameangle?Cantheypairananglewithoneormoreofitsnames?Cantheyfindtwoormorenamesforthesameangleeveniftheangleitselfisnotamongthoseshowing?Whentheyhavefamiliarizedthemselveswiththecardsinthisway,theyarereadytoplaythegame.

1. Theobjectistocollectpairsofcardswhichrepresentthesameangle.Thesemayeitherbeadrawingofananglewithoneofitsnames,ortwonamesforthesameangle.Twodrawingsofthesameangleindifferentpositionsarenotanallowablepair.

2. Thepackisshuffledandturnedfacedown.Thefirstplayerturnsoverthreecards,andifheseesapairhetakesit.

3. Thereafter,eachplayerturnsoverthetopcardfromtheface-downpack,andputsitbesidetheotherface-upcards.Ifheseesapair,hetakesit.Sincethisisunlikelytohappenateveryturn,thenumberofcardstochoosefromwillgraduallyincrease.

4. Iftheplayerwhoseturnitisseesanunclaimedpair,hemaytakeitbeforetakinghisnormalturnasinstep3.

5. Steps3and4arerepeateduntilallthecardshavebeenturnedover,andeveryplayerhashadachancetotakeanyunclaimedpairs.Thewinneristheplayerwithmostpairs.

6. Anotherroundmaythenbeplayed.

Activity 3 Words from compass bearings [Space 2.7/3]

Anactivityforasmallgroupofchildren,similarto‘DirectionsforWords,’Space2.6/2,butusingcompassbearingsupto360˚insteadofthecompassdirectionsnorth,south,etc.Itthususesafamiliarsettingtointroducethemoredifficult,butmoreaccurate,methodforgivingcompassdirections.Forthispurposeweusejustthemultiplesof30˚,whichgivesachoicefrom12letters.Thisestablishesafull-circleframeworkwithinwhichfurtherdetailcanbedevelopedlater,whenchildrenarelearningtouseprotractors.

Thedirectionsarerepeatedhereforconvenience.

Rules of the game

Space 2.7 Angles as amount of turn; compass bearings (cont.)

340

Materials Foreachplayer: • Agameboardwithlettersplacedontwelvecompassbearings.* • Atwo-sideddirectionscardshowingononesidethetwelvecompassbearings

0˚,30˚,60˚,90˚,...300˚,330˚,andontheothersideasinglearrowforindicatingnorth.*

• Paperandpencil. Forthegroup: • Twosetsofwordcards,easierand harder.* *Providedinthephotomasters

What they do Stage (a) 1. Eachplayerhasanactivityboard,

and puts his directions card on the middle with the north direction pointingawayfromhim.Tobeginwith,theyallhavethesideshowingallthedirectionsfaceup,withtheNORTHarrowpointingawayfromthem.Inthiscase,itwillbepointingtotheletterR.

2. Oneplayeralsohasoneofthesetsofwordcards.Hepicksoneoftheseatrandom,whichhelooksathimself but does not let the others see.

3. Hethencallsoutthedirectionsforthelettersinorder.E.g.,“Firstletter,330˚.”Theotherplayersallwritethis.“Nextletter,zerodegrees.”Theotherplayerswritethis;andsoonuntilthewordiscomplete.

4. Thecallerthenshowsthewordtotheotherplayers.Ifsomedonothavethesameword,herepeatsthedirectionsgiveninstep3whiletheotherscheckwhattheyhavewrittentofindwhetheritwastheyorthecallerwhomadethemistake.

5. Anotherplayerthentakesovertheroleofcaller,andsteps2to4arerepeated. Stage (b) ThisisplayedinthesamewayasStage(a)exceptthattheyturnovertheircards,so

thatthesideshowingistheoneonwhichonlynorthismarked.Thecaller(only)mayifnecessarytakealookattheothersidetoensurethathiscallsarecorrect,butwithoutlettinganyoneelsesee.

Variation Thisgoesbestwhenusingtheharderpack.Ifaplayerthinksheknowstheword,hemayholdupahandandsaythenextdirection(nottheworditself).Thecallerwillsaywhetherornotthisiscorrect.Onlyoneguessperplayerisalloweduntilallhavehadachancetoguess.

Notes (i) Theeasierpackcontainsonlywordsoffourorfiveletters,includingsomeplurals.Theharderpackcontainsonlywordsoffivelettersupwards.

(ii) Youmightwishtolookthroughthewordcardsandremoveanywhichyouthinkthechildrenmightnotknow,e.g.,‘arbitrate.’Alternatively,youmightthinkitusefultoleavethesein,toenlargetheirvocabularies.

Space 2.7 Angles as amount of turn; compass bearings (cont.)

One side of two-sided directions card(on game board)

210

0

90270

180

330 30

120240

RB

O

G

IS

H

T

Stage (a)

150

300 60

N A

E

D

L

341

Activity 4 escape to freedom [Space 2.7/4]

Anactivityforasmallgroup,workinginpairs.Itspurposeistointroducetheuseofcompassdirectionsforfindingtheway.Stage(a)shouldnotbehardforchildrenwhohavedonetheearlieractivitiesinthistopic.Stage(b)ismoredifficult.

Materials Foreachchild: • Anactivitysheetrepresentingamaze.(Figure40containsareproductionofa

mazedesignedbyJodiePalmer,oneofthefield-testschoolchildren.)* • Anon-permanentmarker. ForStage(b),also: • Pencilanderaser. • Ballpointpen. • Clearacetatefilefolder. Forthegroup: • Dampspongeorequivalent. *Photomastersareprovidedforthreedifferentmazes,andalsoformakingone’s

ownmazes.Theseneedtobeprovidedinpairsalike,andlaminatedsothatlinesdrawnwiththenon-permanentmarkerscanbeerasedafterwards.

What they do Stage (a) 1. Eachpairofchildrenreceivestwoactivitysheetswiththesamemazeonit. 2. Theyreadthedirections,whichreadasfollows.

You have been imprisoned in a maze by a cruel tyrant. However, concealed in your clothing you have a compass. Moreover, a freedom fighter has scratched numbers on the floor of every cell, and these will help you to escape if you can interpret them. But if you get off the path, you may find wrong directions in some of the cells, so be careful. Good luck.

3. Explainthatsincetheyarelookingatapictureofthewholemaze,itiseasyforthemtoseethewayout.Butweareaskingthemtoimaginethattheyarealwaysinsideoneofthecells,wheretheywouldonlybeabletolookthroughtheopeningintotheadjacentcellandnofurther,untiltheyareoutofthemaze.

4. Explainthatpartnershavethesamemazes.Sothattheycancomparetheroutestheyhavetaken,theyaretomarktheseastheygowiththeirnon-permanentmarkers.

5. Whentheyhavefoundthewayout,eachcompareswithhispartner.Sincethereisonlyonecorrectwayoutforeachmaze,thisprovidesacheckthattheyhavefollowedthedirectionscorrectly.

6. Aftererasingtheroutes,theymayexchangemazesandrepeatsteps1to5.

Stage (b) 1. Theymaynowbereadytodesigntheirownmazes.Thesuggestionsbelowoffer

onewayforsettingaboutthistask.Alternatively,youmaywishtoleavetheplanningtothem.

2. EachchildreceivesacopyofthesheetMAKEYOUROWNMAZEinaclearacetatefilefolder.

Space 2.7 Angles as amount of turn; compass bearings (cont.)

342

Figure 40 A pupil-designed maze

Space 2.7 Angles as amount of turn; compass bearings (cont.)

343

3. Heusesthistomarkhisstartingpointandexitrouteontheacetate,usingthenon-permanentmarker.

4. Heclosesallopeningswhichwouldallowshort-cuts.Incorrectpathsshouldnotbeprevented.

6. Next,hewritesinthenumbersforcompassbearingswhichshowthecorrectroute.

7. Thiswouldbeagoodstagetoexchangemazeswithapartnerforcheckingsofar.(Partnerswillnotinthiscasebeworkingonthesamemaze.)

8. Thesenumbers,ifcorrect,aretransferredinpenciltothepapercopy,andothernumbersareenteredinthecellswhichdonotformpartoftheroute.

9. Theacetatecoversarenowwipedclear,andthemazestriedoutbysomeoneelseinthegroup.(Ifsparecoversareavailable,themazemaybetransferredtoacleancoverandtheoriginaltemporarilykeptforreferenceuntilthemazeisfinalized.)

10. Ifsatisfactory,thepencillednumbersarerewritteninballpointpen,andthemazemaythenbeaddedtotheschool’scollection.

Thefirstthreeactivitiesbuildontopic6tointroduceanglesasrepresentingturnsanddirections,anddegreeswherebythesemaybespecifiedwithcloseraccuracy.Activity3inparticularisdesignedtogivechildrenfluencyinrelatinganglestodirectionsoverthefullrangeof360˚ratherthanthemorefrequentlytaughtrange0˚-180˚.Thelatterisusuallysufficientfortriangles,otherpolygons,andtheshapeaspectofanglesingeneral,sincereflexanglesarenotfrequentlyencountered.However,allanglesinthefullcircleareequallyimportantinthecontextofdirections. Activity4relatescompassbearingstomovementsinanimaginarysituation.Inthenexttopic,weshallapplytheseconceptstoareal-lifecontext.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 2.7 Angles as amount of turn; compass bearings (cont.)

Discussion of activities

Escape to freedom [Space 2.7/4]

344

Space 2.8 DIRECTIONS AND LOCATIONS

Concepts (i)Adirectionaspointingtowardsalocation. (ii)Backbearings,astheoppositedirectiontoagivendirection. (iii)Theintersectionoftwodirectionsasdeterminingalocation.

Abilities (i)Touseamapandcompasstogotoachosenlocation. (ii)Tofindone’swaybacktowhereonestarted. (iii)Tousemapandcompasstofindoutwhereoneis.

InSpace1,angleswereseenasanaspectofshape.Inthepresentnetwork,theyhavebeenconsideredasadifferencebetweentwodirections,andastheamountofturnfromonedirectiontoanother.Bychoosingafixeddirectionasastartingdirection,usuallynorth,otherdirectionscanbespecifiedasamountofturnfromthisstartingdirection.Inthepresenttopic,directionsarefoundtobeanimportantwayofdeterminingalocation,therebyintroducinganddevelopingfurtherrelationshipsamongangles,lines,andpoints.

Activity 1 introduction to back bearings [Space 2.8/1]

Ateacher-ledactivityforasmallgroupofchildren.Thepurposeofthisandthenexttwoactivitiesistoteachthemhowtouseacompassforfindingtheirwaybacktowheretheycamefrom.Thisknowledgemightsavetheirlives,oneday.

Materials Foreachchild: • Pencil,paper,andruler. • Compassbearingscard* *Providedinthephotomasters Suggested Stage (a) introduction 1. Askthechildrentolookattheircards,

which show the same set of compass bearingsastheyhavealreadyusedinSpace2.7/3.“Wearegoingtotalkaboutdirectionsandtheiropposites.”

2. “Supposethatyouarepointinginacompassdirectionof60˚,andyoumakeahalfturnsoastopointinexactlytheoppositedirection,whatwouldthenewdirectionbe?”(240˚)“Inotherwords,theoppositedirectionof60˚is240˚.Youcancheckthisbylayingyourruleralongthe60˚line.Thisshowsthetwooppositedirections.”

Discussion of concepts

Starting Opposite direction direction 0˚ 180˚ 30˚ 210˚ 60˚ ... ... 180˚

210

0

90270

180

330 30

120240

RB

O

G

IS

H

T

Stage (a)

150

300 60

N A

E

D

L

345

3. Letthemrepeatthiswithotheranglesbothlessthanandgreaterthan180˚. 4. Askthemtomakeatableofdirectionsandtheiropposites,startingasshown

belowthecompassbearingscard.Tostartwith,theyaretostopat180˚. 5. Askthemtolookattheirtablesandtrytofindapattern.Hint:howmuchgreater

iseachoftheoppositedirectionsthanitsstartingdirection?Isthisamountalwaysthesame?Ifso,why?(Always180˚greater.Topointintheoppositedirectionrequiresahalfturn,whichis180˚.)

6. Nowaskthemtocompletetheirtable,upto330˚,asbeforewiththehelpoftheircompasscards.“Isitstillthesamepattern?”(No.Forthese,wehavetosubtract180˚.)Whyisthis?(Twowaysofexplaining.Ifweadd180˚asbefore,thistakesuspast360˚,sowehavetoconvertbysubtracting360.Adding180andthensubtracting360isequivalenttosubtracting180.Asimplerexplanationistoturnanti-clockwise.)

7. Explainthattheseoppositedirectionsarecalledbackbearings.E.g.,thebackbearingfor60˚is240˚.Askforotherexamplesusingthisnameforthem.

8. “Whydoweneedtobeabletogetbackbearingsbycalculation?”(Calculationismorereliableforintermediatedirectionslike72˚and249˚,forexample.Also,wemaynothaveacompasscardwithus.Sometimeswecanreaditoffthecompassitself,butthisisnotalwayseasy.Finally,itisalwaysgoodtohaveawayofcheckingourresults.)

Activity 2 get back safely [Space 2.8/2]

Anactivityforasmallgroupofchildren,workinginpairs.Stage(a)introducestheuseofbackbearingsinanimaginarysituation.Stage(b)thenappliestheknowledgeinanoutdoorsituationwhichisalittleclosertothereal-lifeapplication.

Materials Forthegroup: • Acompass.* Foreachchild: • Acopyofthemap,asillustratedinFigure41.† • Transparentcompassrose.†‡ • Pencilandpaper. • Eraser. • Ruler. *ThisisnotneededuntilStage(b).Itwouldbegoodifaprismaticcompasscould

bemadeavailable. †Providedinthephotomasters ‡Thisneedstobecopiedontotransparentfilm,suchasthatusedforoverhead

transparencies,andthencutoutcarefully.Alternatively,a360˚protractormaybeused.

Note Atthisstage,Ithoughtitbetternottointroducethedifferencebetweentruenorthandmagneticnorth,whichisanadditionalcomplication.IfinStage(b)bearingsusingmagneticnorthareusedonbothoutwardandreturnjourneys,thedifferenceof180˚stillappliesandthebackbearingthuscalculatedwilltakethembacktowheretheystarted.Thiswillbedealtwithinthenextactivity,“Wherearewe?”

Space 2.8 Directions and locations (cont.)

346

Figure 41 ‘Get back safely’ map

Space 2.8 Directions and locations (cont.)

347

What they do Stage (a) 1. Tostartwith,theyallwritethreeheadings:

Destination Outwardbearing Backbearing

2. Eachofthemthenchooses,inturnwithherpartner,twoormoredifferentdestinations,andenterstheseonhertable.Theseshouldincludesomelocationsontheeasternsideofthemap,andsomeonthewestandnorthwest.

3. Theythenfillintheoutwardbearings(only)bylayingarulerbetweenthecampareaandthedestination,andputtingthecompassroseontopwiththecentreinthemiddleofthecamparea.Itispossibletogetthenorthdirectionaccuratetowithinoneortwodegreesbymakingitperpendiculartothetopedgeofthemap.

4. Theythencalculatethebackbearingwhichwillgetthembacksafelyfromtheirdestinations,andaddthesetotheirtables.TheseresultsarethencheckedusingthecompassroseinthewaythecompassbearingscardwasusedinActivity1.

5. Theyexchangepaperswiththeirpartners,andcheckeachother’soutwardandreturnbearings.Atthisstage,therulerandcompassrosemethodisusedforallthebearings.Sinceallofthelocationsarelargerthanapoint,somevariationinthefigureisallowable.Thequestionis,willtheoutwardbearinggetthemtothecorrectlocation,andwillthebackbearinggetthemsafelyback?Bothhavetobecorrect,withreasonableaccuracy,ofcourse.

6. Anydisagreementsarethendiscussed,afterwhichtheprocessmayberepeateduntilallthelocationshavebeenused.

7. TheyshouldthenbereadyforStage(b).

Stage (b) Asandwhenopportunitypermits,itwillbevaluableforthemtousethemethod

learnedinStage(a)forsmalldistancesoutdoorswithacompass.Thiscouldbe,e.g.,inaplayground,playingfield,parkorotheropenarea.Atthisstagenomapisnecessary.Theysimplychooseavisibledestination,finditsbearingwiththecompass,calculatethebackbearingandwriteitdown.Theythenwalktothechosenlocation,andusethecompassandbackbearingtopointthembacktowheretheystarted.Theycanthenseewhethertheywouldindeedgetbacksafelyif(say)aheavymistdescended.Atthisstage,theyareinnorealdangerofgettinglost;buttheyareconsolidatingtheknowledgewhichwillgetthemhomeinfuturesituationswhentheirdestinationsarenolongervisible.

Activity 3 Where are we? [Space 2.8/3]

Anactivityforasmallgroupofchildren.Thisisafurtherapplicationofbackbearings.InActivity2,theylearnedhowtousebackbearingstore-tracetheirstepsfromaknowndestinationtowheretheycamefrom.Itisamethodfornotgettinglost.Inthepresentactivity,theylearnhowtofindwheretheyareiftheydogetlost.

Space 2.8 Directions and locations (cont.)

348

Space 2.8 Directions and locations (cont.)

Suggested introduction for the group as a whole

Name Landmark Bearing Back bearing Location Alice Flag 56˚ Radio mast 296˚ Brian Obsvn hut 232˚ Radio mast 330˚

Materials Foreachchild: • Acopyofthelocationsmap.* • Worksheet.* • Atransparentcompassrose.† • Ruler. • Pencilandpaper. • Eraser. *Providedinthephotomasters.SeealsoFigure42. †Asusedinthepreviousactivity.

1. Theybeginbylookingattheirlocationmaps,andnotingthatthesearelikethosewhichtheyusedinActivity2,withjusttwodifferences.First,thetreesarereplacedbyanumberofcircles,representinglocations.Eachlocationislabelledwithaletterortwoletters.Second,threemoreobjectsareshownwhicharevisiblefromnearlyanywhere.Thesearearadiotransmittermast,anobservationhutattheendofahighridge,andatallflagstaffneartheboathouse.Explainthatinthisactivity,theywillimaginethattheyarelostsomewhereintheareashownbythemap,andwilllearnhowtouseobservationsoftheselandmarkstofindwheretheyare.

2. Nexttheylookattheirworksheets,thetoppartofwhichisillustratedhere.

3. Aliceislost.Wisely,shehasbroughtwithheramapandcompass,andknowshowtousebackbearingstofindwheresheis.Thisiswhatshedoes.

4. First,shelooksaroundforsomeprominentlandmarkswhicharealsoshownonhermap.Sheseestheflagbytheboathouse,andtheradiomast.Soshetakesbearingsonthese,andwritesthemdown.(Yes,shehaspencilandpapertoo!)Thesearetheonesshownontheworksheet.

5. Nextshecalculatesthebackbearingsfromthese,andwritesthesedowntoo.Theotherchildrenalsocalculatethebackbearingsbythemethodtheyalreadyknow,andwritethemintotheirworksheets.

Name Landmark Bearing Back bearing Location Alice Flag 56˚ 236˚ Radio mast 296˚ 116˚ Brian Obsvn hut 232˚ Radio mast 330˚

349

Figure 42 ‘Where are we?’ locations map

Space 2.8 Directions and locations (cont.)

350

6. Alicenowknowsthatherdirectionfromtheflagis236˚.Ifshedrawsalineonthemapfromtheflaginthisdirection,sheissomewhereonit.Likewise,ifshedrawsalinefromtheradiomastinthecompassdirectiondirection116˚,sheissomewhereonthislinealso.Sowherethetwolinesintersectonthemapiswheresheis.Aliceisnolongerlost.

7. ThechildrenshouldplotAlice’spositionontheirownlocationmapsandcheckthattheygetthesamelocation.(ItislocationAP.)

8. TheymaythendolikewisetofindwhereBrianis,havingfirsterasedthepreviouspairoflines.(BrianisatlocationAL.)

9. Theyshouldnowbereadytoconsolidatethis,workingtogetherinpairs.

What they do 1. Eachchoosesoneofthecirclesrepresentinglocationsonthemap,andwritesitsidentifyingletter(s)onthebackoftheirpaperswithoutlettingtheirpartnersseeit(them).Thepapersarethenturnedoveragain.

2. Theyeachentertheirownnamesinthenextspaceintheleft-handcolumn.Theythenusetheirtransparentcompassrosestofindthebearingsfromtheirchosenlocationsoftwolandmarks,andentertheseintheirtables.

3. Whenbothhavecompletedthis,theycopyeachother’sentriesintotheirowntables.

4. Whenbothhavecopiedeachother’sentries,theyfindeachother’slocationsbythesamemethodasbefore.Ifthesearewhattheirpartnerhaswrittenonthebackofherpaper,congratulationstobothofthem.Ifnot,theycheckeachother’sworktofindwhohasmadeamistake,andwhere.

Notes (i) Greateraccuracymaybeobtainedbytakingbearingsofthreelandmarks.Thethreelinesofthebackbearingswillnowprobablyencloseasmalltriangle,andthelikeliestlocationisatthecentreofthis.

(ii) Theflagstaffisshownonthemapasthoughitwasvertical,forgreaterrealism:sopupilswillneedtoagreewhatpartofitshouldbeusedforbearings.IntheillustrationIhaveusedtheflag,sincethisislikelytobemostvisiblefromafar.

Activity 4 true north and magnetic north [Space 2.8/4]

ThisissomefurtherinformationwhichshouldbeknownbeforeusingthemethodofActivity3inanoutdoorsituation.Itsimportanceincreaseswiththedistanceinvolved.Forexample,atadistanceof1kmfromtheobservationpoint,thedistancebetweentruenorthandmagneticnorthis87metres,foramagneticdeviationof5˚.

1. Northonamagneticcompass,andnorthonamap,arenearlybutnotexactlythesame.

2. Mapsusetruenorth,whichisthedirectionoftheearth’snorthpole.Theeasiestwaytofindthisdirectionisbythepolestar,whichshowstruenorth.

3. Manymapsalsostatethemagneticdeviation,i.e.,thedifferencebetweentruenorthandmagneticnorth.Thisvariesfordifferentpartsoftheworld,andalsochangesbyasmallamountannually.

Suggested outline for the discussion

Space 2.8 Directions and locations (cont.)

351

Atypicalvalueis5˚west,whichmeans that magnetic north is 5 degreeswestoftruenorth.Atruebearingof,say,30˚isthereforeequivalenttoamagneticbearingof(inthiscase)35˚.Somagneticbearingsareconvertedtotruebearingsby(inthislocation)subtracting5˚.

4. However,somecompasseshavebuilt-inwaysforconvertingautomaticallytobearingsbasedontruenorth.Theseareespeciallyusefulunderdifficultorstressfulconditions.

5. ThedifferencebetweentruenorthandmagneticnorthdoesnotmatterforStage(b)ofActivity2,forseveralreasons.First,thedistancesaresmall.Second,theyarenotusingmaps.Third,providedthatbearingsusingmagneticnorthareusedforbothoutwardandreturnjourneys,thedifferenceisstill180˚andthebackbearingthuscalculatedwilltakethembacktowheretheystarted.

Insomepartsoftheworld,itisarguablethattheknowledgeandabilitiesdescribedinthistopicareasimportantsurvivalskillsaslearningtoswim.Themethodforfindingone’slocationdescribedinActivity3hasbeenastandardmethodforcoastalnavigationeversincetheintroductionofthemagneticcompass.Usingmoresophisticatedtechnologysuchasradiobeaconsanddirectionfindingapparatus,thesameprincipleisusedworldwidebynavigatorsofshipsandaircraft.

OBSERVE AND LISTEN REFLECT DISCUSS

Space 2.8 Directions and locations (cont.)

Discussion of activities

truen

5˚ 30˚

magneticn

Where are we? [Space 2.8/3]

352

Space 2.9 ROTATIONS OF TWO-DIMENSIONAL FIGURES

Concepts (i) Rotationofatwodimensionalobjectinitsownplane. (ii) Centreandangleofrotationasbeingthetwofactorswhichdeterminearotation.

Abilities (i)Toapplyagivenrotationtoachosentwo-dimensionalobject. (ii)Tochooseacentreofrotationwhichwillachieveaparticularresult. (iii)Torecognizewhenafigurehasrotationalsymmetry,andtostatethenumberand

anglesofrotation.

Attentionwasfirstfocusedonthedistinctionbetweenlinearandrotarymotionintopic3.Anylinearmotion(translations,slides)islinkedwithadirection,sothiswasthesubjectoftopics4and5.Therelatedaspectsofanglewerealsoconsideredintheseoversevenactivities.Wenowconsideranotherfeatureofrotation.Againwestartwithaneverydayconcept,andbringitintoamathematicalcontext. Theeffectofarotationwhenappliedtoatwo-dimensionalfigureisdeterminedbytwofactors:itscentreandtheangleofrotation,whichmayvaryinamountanddirection(clockwiseorcounterclockwise).Notethatthecentreofrotationmaybewithinoroutsidethefigure,oronitsboundary. Rotationalsymmetryisanotherspatialexampleofamathematicalpattern.Thisfurtherhelpstointegratethenumericalandspatialaspectsofmathematics.

Activity 1 Centre and angle of rotation [Space 2.9/1]

Ateacher-leddiscussionforasmallgroup.Itspurposeistofocusattentiononthetwofactorswhichdetermineaparticularrotation,namely,‘thecentreofrotation’and‘theangleofrotation.’

Materials • Baseboardsandcutoutshapes,one foreachpairofchildren.* • Cocktailstick,toothpick,orthelike. *Providedinthephotomasters

1. Putoutabaseboardwithoneofthecutoutshapesinthematchingoutline.

2. Showhow,bypressingdownthroughtheholeinoneofthedots,thefigurecan be rotated about this point as centre.

3. Demonstraterotationabouttheotherdotascentre.

4. Thefigurecan,ofcourse,berotatedaboutanypointwhichwecankeepstill.Thedotsarejusttoshowthecentreofrotationifthestickisremoved,andtheoutlineshowsthepositionofthefigurebeforerotations.

Discussion of concepts

Suggested outline for the discussion

353

5. Wecanalsorotateafigureaboutapointoutsideit.Inthiscasewehavetousethefigureonthebaseboarditself.

6. Usetwocutoutstoshowtheeffectsofaquarter-turn(90˚)rotationabouttwodifferentvertices.Whatstaysthesame?(Thenewdirectionsofallthelines.)Andwhatisdifferent?(Thepositionofallthepoints.)

7. Istheabovealwaysthecase,forthesame amount of turn about different centres?

8. Letthemhaveaboardforeachpairofchildren,withtwocutouts;encouragethemtoexperimentfurther.Forexample,theyshouldtryusingonecutouttoshowtheeffectofa90˚rotationclockwise,leavethisinposition,andthen,usingthesamecentreofrotationwithanothercutout,comparetheeffectofa270˚rotationcounterclockwise.Likewisewithhalfturns.Cantheyfindanyotherequivalentpairs?

9. Allowthemtoexperiment,untiltheyhaveunderstoodtherespectivecontributionsofthecentreandangleofrotationonthefinalpositionandaresomewhatfamiliarwithusingdifferentanglesofrotation.TheywillthenbereadyforActivity2.

Activity 2 Walking the planks [Space 2.9/2]

Agamefortwoorthreechildren:possiblyfour,butabovethreeitisprobablybettertohaveasecondsetofmaterials.ItspurposeistoconsolidatetheconceptsformedinActivity1byusingtheminanewsituation.

Materials • Gameboard,seeFigure43.* • Packofanglecards.* • Planks,oneforeachplayer.† *Providedinthephotomasters †Madefrompopsiclesticks.Theyshouldhavenotches,orotherdistinguishing

markstotellthemapart.

Space 2.9 Rotations of two-dimensional figures (cont.)

Figure 43 ‘Walking the planks’ game board

354

Rules of the game 1. Thegameboardrepresentsaswampyareabetweentwofirmbanks.Thereused

tobeseveralplankbridgesacross,butthesehaverottedaway.Someofthesupportingpostsarestillthere,andtheseareshownbyblackcircles.Thelighterareaaroundtheserepresentstheconcreteinwhichthepostsareembedded.

2. Theaimistocrosstheswampwiththehelpofsingleplankswhicheachplayercarrieswithhimashegoes.Thesecanrestonlyonthepoststhemselves,butplayerscanstandontheconcretesurrounds,providedthattheyhavegoodbalance.Theyneedtodothiswhilemovingtheirplanksforwardtothenextposition.Itisnotpossibletohavetwoplanksrestingonthesamepole.

3. Thegamebeginswitheachplayerinturnputtingdownhisplankwithoneendonthebankandtheotheronanyunoccupiedpost,pointingdirectlytowardsthefarshore.

4. Eachnewpositionofaplankisachievedbyarotation.Thecentreofrotationmaybechosenbytheplayer,but the angle of rotation is determined asfollows.(ThisisthesameasinIslandCruising[Space2.6/3],usingdifferentcards.)

5. Theanglespackisshuffledandputfacedownonthetable.Thetopthreecards are now turned face up and put separatelybesidethepack.

6. Theplayerwhoseturnitismaytakejustoneoftheface-upcards,androtatehisplankthroughthatangle,clockwiseorcounterclockwise.Ifnoneoftheseisofuse,theturnpassestothenextplayer.

7. Ifaplayertakesacard,hereplacesitbyturningoverthetopcardfromtheface-downpacksothattherearealwaysatleastthreeface-upcardstochoosefrom.

8. Ifawholeroundgoeswithoutanyplayertakingacardfromthethreeface-upones,theseareputasideandthreenewonesareturnedover.Ifthewholepackisusedbeforealltheplayershavecrossed,allthecardsarereturnedandthepackisshuffledreadyforfurtheruseasabove.

9. Thewinneristhefirstacross,butplaycontinuesuntilallhavereachedthefarshore.

Starting position

After 180˚ rotationin either direction

After 225˚ rotation clockwise or 135˚ rotation counterclockwise

Space 2.9 Rotations of two-dimensional figures (cont.)

355

Activity 3 introduction to rotational symmetry [Space 2.9/3]

Ateacher-leddiscussionforasmallgroup.Itspurposeisindicatedbythetitle.

Materials • Examplescards,setsAandB.* • AsmallpieceofPlasticine,Blu-tak,orthelike. *Providedinthephotomasters.SetAisusedforStage(a)ofthepresentactivity,

andsetBforStage(b)ofthepresentactivityandalsoforActivities4,5,and6.

Stage (a) 1. Putexamplecards1and2sideby

side.Explainthatthesearetwoexactcopiesofthesamediagram,madebyacomputer.

2. Rotatetheright-handonethroughaquarterturnclockwise,andaskwhattheynotice.(Theystilllookthesame.)Explainthatifafigurehasthisproperty,wesaythatithasrotational symmetry.

3. Cantheyseeanyotherangleforwhichthiswouldbetrue?(Yes,ahalfturn.)

4. Itisgettinghardtoseehowmuchithasbeenturned,soletusstartagainand put little blobs of plasticine to showwhereitstarts.

5. Nowtheybothstartinthesameposition,andaswerotatetheright-handfigurewecanseewhatturnhasbeenmade.

6. Whatrotationalsymmetrieshavewefoundsofar?(Quarter,half,andthreequarterturns.)Andindegrees?(90˚,180˚,270˚)

7. Havewegotthemallnow?(Andiftheyanswer“Yes”...)Shouldweincludeafullturn?(Itcertainlylooksexactlythesame,buteveryfigurehasthissymmetry,evenasquiggle.Sowe

call this one trivial.Thisfigurehasjustthreenon-trivialsymmetries,forrotationsof90˚,180˚,and270˚,whichisthesameasquarter,half,andthree-quarterturns.)

8. Nowputoutthenexttwocards,andasksimilarquestionsforthisfigure.Testwiththeplasticineblobifdesired.(Ithasjustonenon-trivialsymmetry,forrotationofhalfaturn,or180˚.)

Space 2.9 Rotations of two-dimensional figures (cont.)

Suggested outline for the discussion

356

9. Andthis?(Twonon-trivialsymmetries,for120˚and240˚.)

10. ThePlasticinewillbeusefulforthisone,whichhasfournon-trivialsymmetries,forrotationsof72˚,144˚,216˚,and288˚.(Theymightbegladofpencilandpaperhere,oracalculator!)

Stage (b) 1. ReplacethesetAcardswithsetB.Spreadoutsomeofthesefaceupwards,and

invitethechildrentosortthemintogroupswiththesamenumberofrotationalsymmetries.(Thepackdoesinfacthavefivecardsforeachof0,1,2,3,4,and5rotationalsymmetries:total,30cards.)

2. Invitethemtosayhowmanyrotationalsymmetrieseachgrouphas,andalsowhattheanglesare.

3. TheyshouldthenbereadyforActivity4.

Activity 4 match and mix: rotational symmetries [Space 2.9/4]

Agamefor2to5players,beinganothermemberofthe‘MatchandMix’family.ItspurposeistoconsolidatetheconceptsformedinActivity1,usingavarietyofexamples.Therulesarethesameasfortheother‘MatchandMix’games,butarerepeatedhereforconvenience.

Materials • CardssetB,asusedinStage(b)ofActivity3. • Matchandmixcard.* • Scoringremindercard.* *Providedinthephotomasters

1. Thecardsarespreadoutfacedownwardsinthemiddleofthetable. 2. TheMATCHandMIXcardisputwhereverconvenient. 3. Eachplayertakes5cards(if2playersonly,theytake7cardseach).

Alternativelythecardsmaybedealtintheusualway. 4. Theycollecttheircardsinapilefacedownwards. 5. Playersinturnlookatthetopcardintheirpiles,andputcardsdownnextto

cardsalreadythere(afterthefirst)accordingtothefollowingthreerules. (i)Cardsmustmatchorbedifferent,accordingastheyareputnexttoeach

otherinthe‘match’or‘mix’directions.Here,‘match’or‘mix’referstothenumberofrotationalsymmetries,rememberingthatinsomecasesthisnumberiszero.

(ii)Notmorethan3cardsmaybeputtogetherineitherdirection. (iii)Theremaynotbetwothreesnexttoeachother.Thisistokeepanopen

pattern,withasmanygrowingpointsaspossible.

Rules of the game

Space 2.9 Rotations of two-dimensional figures (cont.)

357

6. Acardwhichcannotbeplayedisreplacedatthebottomofthepile. Atypicalarrangement(usingA,B,C...fordifferentkindsofrotationalsymmetry):

Noneoftheseisallowed:

7. Scoringisasfollows: 1pointforcompletingaroworcolumnofthree,withanextrapointforcorrectly

namingalltheanglesofrotationalsymmetry. 2pointsforputtingdownacardwhichsimultaneouslymatchesonewayand

mixestheotherway. 2pointsforbeingthefirst‘out’;1pointforbeingthesecond‘out.’ (Soitwouldbepossibletoscoreupto7pointsinasingleturn.) 8. Playcontinuesuntilallplayershaveputdownalltheircards. 9. Anotherroundmaythenbeplayed.

Activity 5 match and mix (line symmetry) [Space 2.9/5]

SincemanyofthecardsusedinActivity4alsohavelinesymmetries,wehavehereagoodopportunitytoreviewandconsolidatethisconceptwhichwasfirstintroducedintopic2.Italsoprovidessomewhatharderexamplesthanwereusedthen.

Materials • ThesameasforActivity4,withaminordifferenceinthescoringremindercard.* • Somethingwithwhichtoindicatetheaxesofsymmetry,suchasacocktailstick,

astraightpieceofwire,astraightedgeofclearplastic. *Providedinthephotomasters

ThesameasforActivity4,exceptthatinthiscasetheextrapointisscoredforcorrectlyshowingtheaxesofsymmetry(mirrorlines).

Havingestablishedtheconceptofanglesasmeasuringamountofturn,wenowapplythisasamathematicaloperation.Operationslikeadditionandsubtractionareappliedtonumbers(amongotherthings).Here,theoperandsaretwo-dimensionalfigures,andtheresultsarethereforemorevisiblethannumericaloperations. Followingafamiliarsequence,Activity1exploresthisnewconceptwithphysicalembodiments,andActivity2thenconsolidatesitinaboardgame.Thelatterunderlinesthedistinctinfluencesontheoutcomeofthecentreandangleofrotation.Activity3combinestheirexistingconceptofsymmetrywiththejust-acquiredconcept,andcomesupwithanewvarietyofsymmetry.Activity4strengthensthisconceptbyhavingthemapplyittoasubstantialnumberofexamples,someofthemquitedifficult.InActivity5,theyremindthemselvesoftheotherkindofsymmetrywhichtheyknow,andlookforthisinthesamefiguresthattheyhavejustusedforrotationalsymmetry. AsIwritethis,havingjustfinishedfieldtestingtheforegoingfiveactivitieswithgirlsandboysofaverageability,Icontinuetomarvelatwhatthehumanchildcanapprehend,givensuitableresources.

OBSERVE AND LISTEN REFLECT DISCUSS

BBBB CCCDDD

AABBCC

ABA

ABCD

M AT C H I X

C A A AB B B D DD B A

Space 2.9 Rotations of two-dimensional figures (cont.)

Rules of the game

Discussion of activities

358

Space 2.10 RELATION BETWEEN REFLECTIONS, ROTATIONS, AND FLIPS

Theterms‘reflection’and‘flip’areoftenusedinterchangeably.Thepurposeofthepresenttopicistoexaminethisassumptionmoreclosely.Thediscussionwhichfollowsisforyourowninformation,andtherearenoactivitiesforthechildren.Ileaveittoyourownjudgementhowmuchofthefollowingyoupassontothem. Amathematicalreflection(unlikethatineverydaylife)takesplaceentirelyintwodimensions.Itwouldbeusefulatthisstagetoreviewthediscussionoftheconceptofreflectiongivenatthebeginningoftopic1inthisnetwork.Fromthis,andespeciallyfromtheSpace2.1/3worksheet,itcanbeseenthattheoriginalfigure,themirrorline,theresultofreflection,andalltheconstructionlinesareinthesameplane,i.e.,theplaneofthepaper. Bothineverydaylifeandmathematically,however,afliptakesplaceinthreedimensions.Ifwestartwithaplayingcardfaceupandflipit,theresultwillbethesamecardfacedown.Andalthoughthestartandtheresultareinthesameplane,namely,thesurfaceofthetable,duringtheactualprocessofflippingthecardhastocomeoutofthatplane.Twistitandturnithowwemay,thereisnowaythataface-downcardcanmoveintoaface-uppositionwhileremainingflatonthetable.Andinthiscase,theresultisquitedifferentfromthatproducedbyareflection. Thisisbecausethecardisopaque.Ifitweretransparent,thenaftertheflipwewouldstillseethesamefigure,butfromtheotherside.Soeverythingwouldbereversed,inthesamewayasinareflection.Inthiscase,althoughtheprocessisdifferentfromthatofareflection,theresultofafliplooksthesame.Andifthefliptakestheformofarotationaboutthemirrorline,thentheresultinglocationwillbethesameastheresultofareflectioninthemirrorline. Ineverydaylifetherearethustwopossibilities,sincethinflatobjectscanbeeitheropaque(likeaplayingcard)ortransparent(likeanoverheadprojectortransparency).Dothesealternativesapplytoamathematicalplanefigure? No.Amathematicalplaneisnotjustthinlikeaplayingcard;ithasnothicknessatall.Soallpointsinaplaneappearonbothitssides.Theresultofflippingamathematicalplanefigureisthereforelikethatofaneverydayflipofatransparentobject,asdescribedabove.Thetwoprocesseshoweverareimportantlydifferent. ItwaswiththeforegoinginmindthatIdidnotintroducetheterm‘flip’intopic1asbeinginterchangeablewiththeterm‘reflection.’Thepurposeofthisdiscussionhasbeentohelpthereadertomakeaninformedpersonalchoiceinthismatter.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of concepts

359

[NuSp 1] The number Track and The number line Powerful support to our thinking about numbers

NuSp 1.7 UNIT INTERVALS: THE NUMBER LINE

Concepts (i) Unit intervals on a line. (ii) The number line.

Ability To use the number line in the same ways as the number track, in preparation for other uses of the number line.

The differences between a number track and a number line are appreciable, and not immediately obvious. The number track is physical, though we may represent it by a diagram. The number line is conceptual – it is a mental object, though we often use diagrams to help us think about it. The number track is finite, whereas the number line is infi-nite. However far we extend a physical track, it has to end somewhere. But in our thoughts, we can think of a number line as going on and on to infinity. On the number line, numbers are represented by points, not spaces; and opera-tions are represented by movements over intervals on the line, to the right for addi-tion and to the left for subtraction. The concept of a unit interval thus replaces that of a unit object. Also, the number line starts at 0, not at 1. For the counting num-bers, and all positive numbers, we use only the right-hand half of the number line, starting at zero and extending indefinitely to the right. For positive and negative numbers we still use 0 for the origin, but now the number line extends indefinitely to the right (positive numbers) and left (negative numbers).

activity 1 Drawing the number line [NuSp 1.7/1]

This is a simple activity for introducing the concept.

Materials • Pencil and paper for each child.

What they do 1. Ask the children to draw a line, as long as will conveniently go on the paper. 2. They mark off equal intervals on it. 3. They number these 0, 1, 2 . . . as in the diagram above. 4. At this stage, the two differences between the number track and the number line

which need to be pointed out are: (i) With the number track, numbers are repre-sented by spaces; with the number line, numbers are represented by points on the line. Though it is helpful to use different marks for tens, fives, and ones, it is the points on the line which represent the numbers. (ii) The number track starts at 1, the number line starts at 0.

Discussion of concepts

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

360

activity 2 Sequences on the number line [NuSp 1.7/2] ‘Sequences on the number track,’NuSp 1.2/1 in Volume 1, may usefully be repeated

here. Smaller markers like the one suggested in the following activity will be needed.

activity 3 Where must the frog land? [NuSp 1.7/3] A game for two players. Its purpose is to introduce the use of the number line for

adding.

Materials • As long a number line as you like (some number lines are provided in the photo-masters).

• A marker representing a frog for each player, occupying as small a base as possible.*

• 1 die 1-6, or 1-10 for athletic frogs. * A short length (about 2 cm) of coloured drinking straw, with a bit of Plasticine on the end, makes a good marker for this and many other activities.

What they do 1. The frogs start at zero. 2. Player A throws the die and tells the frog what number it must hop to. This

is done mentally, using aids such as finger counting if he likes. (To start with, children may use counting on along the number line, but should replace this by mentally adding as soon as they have learned the game.)

3. Player B checks, and if he says “Agree” the frog is allowed to hop. 4. If B does not agree, he says so, and they check. 5. If A has make a mistake, his frog may not hop. 6. Two frogs may be at the same number. 7. They then exchange roles for B’s throw of the die. 8. The winner is the frog which first hops past the end of the line. (The exact

number is not required).

activity 4 hopping backwards [NuSp 1.7/4] The subtraction form of Activity 3, starting at the largest number on the number line

and hopping backwards past zero.

activity 5 Taking [NuSp 1.7/5] Another capture game for two, but quite different from the ‘Capture’ in Volume 1,

NuSp 1.4/4. Its purpose is to give further practice in relating numbers to positions and movements on the number line.

Materials • 1 number line 0-20.* • 3 markers for each player. • 1 die 1-6. * Provided in the photomasters

NuSp 1.7 Unit intervals: the number line (cont.)

361

What they do Form (a) 1. The markers begin at zero. 2. The die is thrown alternately, and according to the number thrown a player may

jump his marker forward that interval on the line. 3. A piece which is jumped over is taken, and removed from the board for the rest

of the game. 4. An occupied point may not be jumped onto. 5. A player does not have to move at all if he doesn’t want to. (We introduced this

rule when we found that starting throws of low numbers were likely to result in the piece being taken next throw, with no room for manoeuvre.)

6. The winner is the player who gets the largest number of pieces past 20. (It is not necessary to throw the exact number.)

Form (b) This may also be played as a subtraction game, backwards from 20.

activity 6 a race through a maze [NuSp 1.7/6] This is a board game for up to 3 players. Its purpose is to bring out the correspond-

ence between the relationships smaller/larger number and left/right hand on the number line.

Materials • Board, see Figure 44.* • Number line 1-20 * • Number cards 1-20 * * Provided in the photomasters For each player: • 1 coloured pointer for the number line. • 1 marker to take through the maze.

What they do 1. The pack of number cards is shuffled and put face down. 2. In turn, each player turns over the top card and puts it face up, starting a new

pile. He may then use this number to position his pointer on the number line, or decide not to (since an extreme left or right position is not helpful). In that case he repeats this step at his next move.

3. When he does position his pointer, he also puts his marker at the start of the maze.

4. After taking steps 2 and 3, players at subsequent turns move forward through the maze if they can. The number they turn over determines whether they can or not.

An example will make it clear.

NuSp 1.7 Unit intervals: the number line (cont.)

362

Since 12 is to the left of the blue pointer’s position, player ‘Blue’ can move for-ward if he is at P on the maze, but not if he is at Q. However, the reverse is the case for player ‘Red’, since 12 is to the right of his pointer.

5. There is no limit to the number of players at a given position on the maze. 6. When all the cards are turned over the number pack is shuffled and used again. 7. The winner is the first player to reach the finish.

Note If the children have difficulty in remembering their left and right, you could help with labelled arrows.

Figure 44 A race through a maze

The first four activities are concerned first with introducing the number line, and then with linking it to concepts which are already familiar. Activity 5, ‘Taking,’ is more difficult, since it involves mentally comparing a number of possible moves before deciding which one to make. Activity 6 involves correspondence not between objects, but between two dif-ferent relations. One is the relation of size, between two numbers, and the other is the relation of position, between two points on the number line. Here is another good example of the conceptual complexity of even elementary mathematics. Yet young children manage this without difficulty if we make it possible for them to use their intelligence to the full.

OBSERVE AND LISTEN REFLECT DISCUSS

NuSp 1.7 Unit intervals: the number line (cont.)

Discussion of activities

363

Discussion of concepts

NuSp 1.8 ExTRApOLATION OF THE NUMBER LINE. ExTRApOLATION OF THE COUNTINg NUMBERS.

Concepts (i) The number line as extending further than we could conveniently draw the whole of.

(ii) The counting numbers as continuing further than we could conveniently count starting at zero.

(iii) The number line as having other points and numbers as well as those marked.

Abilities Given satisfactory ‘landmarks’ on the number line, to construct that part of the number line mentally; and thereby to state the numbers of other points on that part of the line.

When we have clearly seen a pattern, we are often able to extrapolate it, i.e., take the pattern further. Here we have two corresponding patterns: the spatial pattern of the number line, and the pattern of the counting numbers. Each of these has patterns within patterns: the 0-9 cycle is repeated anew, starting again at 10, 20, 30, . . . . The 10, 20, 30, . . . 90 cycle is repeated every hundred; and so on. Since we can think of a line as continuing as far as we like, and as taking these patterns along with it, the number line is an excellent mental support for extending the more abstract concepts of the counting numbers.

activity 1 What number is this? (Single starter) [NuSp 1.8/1]

An activity for children playing in pairs. Its purpose is to start them thinking in terms of a number line longer than one of which one could conveniently draw the whole.

Materials For each pair: • Two cards, each showing part of a number line.* • Two pointers.** * See illustration below. Card (a) is used for Stage (a), Card (b) for Stage (b). ** E.g., Toothpicks or cocktail sticks. We have found that a paper clip, with one end

straightened out, also makes a good pointer.

Card (a)

Card (b)

364

What they do Stage (a) 1. Explain that the card shows part of a number line. The arrows at each end

mean that the line continues back to zero on the left, and as far as we like on the right. The cross-lines mark tens, the smallest marks show ones, and the fives are shown by slightly longer marks.

2. Child A begins by pointing to the left-hand mark, and says (e.g.)

“This number is 40.”

She points again, and says (e.g.)

“What number is this?”

3. If child B says “47,” child A would reply “Agree” and they would change roles. If child B says something else, they would need to discuss.

4. Child A need not always name the left-hand point. E.g., she could begin by saying

“This number is 70”

or “This number is 34.”

5. But A must keep to the tens, fives, ones pattern of marking. E.g., she should not say

“This number is 22.”

If she did, B would say “Disagree” and they would need to discuss. Stage (b) This uses card (b), which covers 2 decades. Otherwise it is played as in Stage (a).

NuSp 1.8 Extrapolation of the number line. Extrapolation of the counting numbers. (cont.)

365

activity 2 What number is this? (Double starter) [NuSp 1.8/2]

A continuation of Activity 1, to be played in pairs. This is appreciably harder.

Materials • The same as for Activity 1.

What they do Stage (a) using Card (a). 1. Child A now begins by placing two pointers and saying what number they

represent. The smaller marks may now represent either unit intervals as before, or intervals of 2, 5, or 10. The larger marks and numbers lines must form part of a ‘sensible’ system. This is best shown by examples.

Sensible.

2. Child A then points to another mark on the line, and asks “What number is this?” If they agree, they interchange roles and steps 1 and 2 are repeated. If not, they discuss.

3. If child B cannot think of a sensible system to explain where A has put the point-ers, she may say “Challenge” and A must explain. What is a sensible system is to some extent open to discussion. Such a discussion would in itself be a valu-able part of the activity. My own view is that the larger marks, and cross lines, should represent more important land marks. Thus I would not see the following as sensible.

Stage (b) This uses card (b); otherwise, as in Stage (a). Stage (c) This uses card (b). The only difference between this and Stage (a) is that the inter-

vals used may be 1, 2, 5, 10, or multiples of these such as 20, 50, 100 etc.

NuSp 1.8 Extrapolation of the number line. Extrapolation of the counting numbers. (cont.)

20 40 25 50

10 1009050

50 100

25 502010

10 90

366

activity 3 is there a limit? [NuSp 1.8/3]

A teacher-led discussion with a group of children. Underlying this is the difference between the finite nature of anything we can do physically, and the in-finite possibili-ties of thought.

1. Ask them how long a line they think they could draw. If they respond with an-swers like “100 miles,” emphasize that you mean actually draw. Lead them in this way to realize the practical limitations of a line drawn physically.

2. Next, ask them how big a number do they think they could actually count – meaning, say the number words? Again let them come to realize the practical limitations.

3. Combine these in the number line. Show them a number line (say, 0-20) drawn on paper and ask them how far do they think this number line could be contin-ued?

4. Now ask them: suppose we don’t try to draw it, but just continue it in our imagi-nation? What number would it take us to?

5. If they do not yet realize that there is no longer any limitation, then Activity 4 will help to lead them towards this realization. If they do, this activity will consolidate it.

activity 4 can you think of . . . ? [NuSp 1.8/4]

A game for two teams. I think it needs to be teacher directed. It might be played with the rest of the class as spectators.

Materials • Chalkboard.

What they do Stage (a) 1. Team A writes a very large number, as big as they like. 2. Team B tries to think of a bigger one. 3. The task becomes easy as soon as they realize that all they need to do is to add 1. 4. When they do, they are ready to go on to Stage b. Stage (b) Can you think of a further point? 1. Team B have now to think of a very long number line, and name (give the

number of) its last point. 2. When they have done so, team A tries to name a point beyond this. 3. Again all they need is to name the point one further on.

There is no largest number! There is no furthest point!

Suggested sequence for discussion

NuSp 1.8 Extrapolation of the number line. Extrapolation of the counting numbers. (cont.)

367

Discussion of activities

NuSp 1.8 Extrapolation of the number line. Extrapolation of the counting numbers. (cont.)

Can you think of . . . ? [NuSp 1.8/4]

This topic is a good example of creative imagination. In Activities 1 and 2, our thinking is supported by visual representations; but the meaning of these becomes increasingly a matter of choice. This choice is, however, neither random nor arbi-trary. It has to ‘make sense’ in terms of what we already know about numbers and the number line. In both of these activities we thus have Mode 3 schema building, and schema testing by Mode 3 (consistency with existing knowledge) and Mode 2 (discussion). Activity 2 is suitable only for the more able children. It requires a well developed feeling for number patterns. With very able children, I have used it without giving them any restriction on what the marks may mean, beyond saying that they must form part of a sensible system. In Activities 3 and 4, children are led to realize that the limitations of working with physical materials disappear when we move into the realm of thought. It then becomes all the more important to have an internally regulated orderliness, such as has been developed in Activities 1 and 2.

OBSERVE AND LISTEN REFLECT DISCUSS

368

Discussion of concepts

NuSp 1.9 INTERpOLATION BETwEEN pOINTS. FRACTIONAL NUMBERS (DECIMAL)

Concepts (i) Points marking tenth-parts of a unit interval. (ii) New numbers, corresponding to these points, called decimal fractions.

Ability To identify tenth-parts of unit intervals on a number line and to associate them with appropriate decimal fractions.

There are still a lot of ‘unused’ points on the number line, that is, points which do not yet have a number. And we do not need to go in the direction of infinity to find them – they are in-between the points already in use, and correspond to fractional numbers. In this topic we shall be concerned mainly with decimal fractional num-bers, usually and inaccurately called ‘decimals.’ The relation between a fraction and a fractional number is discussed in Num 7. As long as we know this difference, there is no harm in following the custom of using the word ‘fraction’ for both. The currently established usage of ‘decimal’ is however less acceptable. ‘Decimal’ simply means ‘related to ten’ (Latin, decem). So ‘decimal notation’ means base ten place-value notation as against (e.g.) base two, or binary, place-value notation. It is inaccurate to talk about ‘decimal num-bers’ and ‘binary numbers,’ and very misleading. The number of objects in a set still remains the same if we write this number differently, just as it does if we use a French or German number-name. ‘Decimal fraction’ does, however describe a different kind of number from the counting numbers. It is a fractional number whose denominator is a power of ten: ten, a hundred, a thousand, etc. So

are all decimal fractions, written in ‘fraction bar’ notation. Decimal fractions have the advantage that we can also write them in the place-value notation already in use. So

.1 .7 .08 .47 .305

are decimal fractions written in (decimal) place-value notation. A ‘decimal’ may or may not be a decimal fraction, and a decimal fraction may or may not be written in place-value notation. We do not need to explain all this to the children, but we must try to avoid mud-dling them, which means we must try to clear up a long-standing muddle in our own minds. It is no accident that ‘decimals’ have long been the bane of genera-tions of school children. I suggest as a reasonable compromise that we accept ‘fraction’ with the dual meaning described above, but say and write ‘decimal fraction’ when we mean a fractional number of the kind described; and ‘place-value notation’ for (e.g.) 17, 0.17, 1.7, etc. This, though not identical with general usage, is not in conflict with it, and therefore should not muddle anyone.

, , , ,

1 7 8 47 305 10 10 100 100 1000

369

activity 1 What number is this? (Decimal fractions) [NuSp 1.9/1]

This is introduced as a teacher-led discussion with a group of children after which it continues as in NuSp 1.8/1.

1. Draw, while you talk, an enlarged part of a number line, starting at zero, as shown below. Point halfway between 0 and 1, and ask

“A half” is a good answer at this stage; we have not yet arrived at decimal frac-

tions. Mark this point in pencil, and do similarly for the points which corre-spond to a quarter and three quarters.

2. We now have the unit interval 0-1 divided into four equal parts. Explain that these are useful divisions, but they don’t fit in with the tens pattern of the rest of the number line.

3. So we rub these out (the half one can stay if you like) and mark the interval 0-1 in ten equal parts. We call these ‘tenth-parts.’ (‘Tenth’ by itself has a different meaning, namely an ordinal number in the sequence first, second . . . ninth, tenth . . .) So the number for

we will call for the present ‘6 tenth-parts.’

And the number for

we will call ‘1 and 6 tenth-parts.’ Once they have grasped this, they are ready to repeat “What number is this?” as

in NuSp 1.8/1, using the same diagrams, but with the smaller divisions repre-senting tenth-parts.

NuSp 1.9 Interpolation between points. Fractional numbers (decimal) (cont.)

Suggested sequence for the discussion

0 1 2 3

What number is this?

0 1 2this point

this point

0 1 2

370

NuSp 1.9 Interpolation between points. Fractional numbers (decimal) (cont.)

activity 2 Snail race [NuSp 1.9/2]

A number line game for up to about four children. Its purpose is to provide a simple introduction to fractional points on the number line.

Materials • Snail race board, see Figure 45.* • Markers for each child to represent snails.** • 1 die 1-6. ** These need to take up as little space as possible, as in NuSp 1.7/2 and 3. * Provided in the photomasters

What they do 1. Players in turn throw the die, but move forward at a snail’s pace! Each number thrown represents only that number of tenth-parts.

2. Passing is allowed. 3. Two snails may not occupy the same point. Points are very small. They must

not move if their roll would take them to an occupied point. 4. Players should declare where they have landed, e.g., “1 and 3 tenth-parts.” 5. They must roll the exact number to finish.

Figure 45 Snail race

371

activity 3 Snails and frogs [NuSp 1.9/3]

This game is played like the one before, with the following modifications. Its pur-pose is to make a contrast between whole and fractional numbers.

Materials • Snails and frogs board, see Figure 46.* • 1 die marked 1, 2, 3 only.** • Markers as before. * Provided in the photomasters * * This can be made from a base-10 unit cube.

What they do 1. Normally a snail moves only that number of tenth-parts, as before. 2. However, if a move takes him exactly to a whole number, he is magically trans-

formed into a frog for his next move, and hops the number of whole unit inter-vals shown by the die.

3. The spell is broken when he lands, however, and he must crawl snail-wise for his next moves.

4. To increase his chance of landing on a whole number, a player may decide not to use his throw. A snail may go into his shell and wait for a beneficial throw. Some children may even calculate the odds that this is a good tactic!

5. Passing is allowed, but two snails may not occupy the same position unless this is a whole number.

6. Players should declare where they have landed, e.g., “4 and 3 tenth parts.” They must score the exact number to finish.

NuSp 1.9 Interpolation between points. Fractional numbers (decimal) (cont.)

Figure 46 Snails and frogs

372

Discussion of activities

NuSp 1.9 Interpolation between points. Fractional numbers (decimal) (cont.)

Snails and frogs [NuSp 1.9/3]

Here again we have a clear example of extrapolation – continuing a known pattern into a new situation. This is of a more difficult kind, since a new variety of number is being mentally constructed. In doing this, the visual support of the number line is a useful aid. Mixed numbers have been used from the beginning here because this gives continuity with the established decimal pattern. (This does not apply in Num 7, where fractions other than decimal are encountered). I hope that you will share my belief in the importance of correct vocabulary when developing this new area of mathematics. I have tried to keep it simple as well as accurate. “Tenth-part” may eventually be shortened to “tenth” (etc.) when the meaning is clear from the context – but not until this meaning is well estab-lished. A muddled vocabulary is a hindrance to clear thinking. We correct bad English in our pupils, so we should not ourselves speak ‘bad mathematics.’

OBSERVE AND LISTEN REFLECT DISCUSS

373

NuSp 1.10 ExTRApOLATION OF pLACE-VALUE NOTATION

Concept The use of place-value notation, including the decimal point, for writing mixed num-bers as represented by points on the number line.

Abilities (i) To write in decimal notation the number for a given point on the number line. (ii) To identify the point for a given mixed number.

In the preceding topic, children have made a start in forming the concept of a new kind of number, a decimal fraction, in the particular context of a number line. They also have spoken names for these new numbers. Now they learn how to write them.

activity 1 “how can we write this number?” (headed columns) [NuSp 1.10/1]

A teacher-led discussion with a group of children. Its purpose is to introduce the use of headed columns for writing fractional numbers.

Materials • A number-line at least 30 cm long, using 1 decimetre (10 cm) as the unit interval.* Centimetres will now represent tenth-parts, and millimetres hundredth-parts. This is introduced in two stages.

• Pencil and paper for everyone. • Pointers. * This may conveniently be made from graph paper.

Stage (a) 1. Begin by saying, “We’ve now learned about new kinds of numbers, called

decimal fractions, and we can say their number names. For example, the name of this number is . . . ?” (Here, use a few examples for review, as in NuSp 1.9/1. At this stage use only the centimetre divisions, representing tenth-parts.)

2. “Now we need a way of writing them.” Begin by drawing headed columns, like those already in use for ones, tens, and hundreds. Discuss the pattern: ten times larger for each column to the left, ten times smaller for each column to the right. This suggests a new column for tenth-parts, on the right of the ones column. A double line separates whole numbers and fractions.

3. Draw this, and let the children draw their own. 4. What number is this?

Discussion of concept

H T Ones t-p

1 3

1 2

Suggested sequence for the discussion

374

Ask how we should write the number on the previous page, spoken as “one and three tenth-parts” and achieve agreement that we should write 1 in the ones column, and 3 in the tenth-parts column. The children should say this aloud, and use the pointers to indicate what they mean.

5. The children may now take turns to point to positions on the line, keeping at this stage to the centimetre divisions. The others write the numbers for these points in their headed columns, and compare.

6. Repeat until they are confident. Stage (b) 1. Point to one of the millimetre divisions and ask what they think these represent. 2. So now we need another column.

3. So what number is this?

“One and three tenth-parts, seven hundredth-parts.” It is written

4. The children then take turns to point while the others say and write the numbers in their headed columns.

activity 2 introducing the decimal point [NuSp 1.10/2]

A teacher-led discussion, introducing the use of a decimal point instead of columns.

Materials • Writing materials for the teacher.

Remind the children that for whole numbers like this one:

we can save having to write the columns every time, and write just

as long as we remember that reading from right to left the digits mean ones, tens, hundreds.

H T Ones t-p h-p

1 3

H T Ones t-p h-p

1 3 7

H T Ones

4 8 5

4 8 5

1 2

NuSp 1.10 Extrapolation of place-value notation (cont.)

Suggested sequence for the discussion

375

This only works if we have nothing smaller than ones. As soon as we do, we can’t simply leave out the columns, or these two numbers

would both be written:

We need to know where whole numbers finish and fractional numbers begin. This is emphasized in headed column notation by the double line. So if we want to drop the columns and use place-value notation, all we need to do is show where the dou-ble line would be. A decimal point does this.

We can now write the above as

2.7 meaning 2 ones, 7 tenth-parts, and .27 meaning 2 tenth-parts, 7 hundredth-parts.

These are read aloud as “Two point seven” and “Point two seven.” It is a good idea to write the second of these as 0.27 to emphasize that there are zero whole numbers.

activity 3 Pointing and writing [NuSp 1.10/3]

A game for a small group. Its purpose is to practise and test their use of place-value notation for decimal fractions, in conjunction with the number line.

Materials • Number line marked in whole numbers, tenth-parts, and hundredth-parts.* • 2 packs of assorted number cards.*† • Pointer. • Pencil and paper for each child. * Provided in the photomasters † Mixed numbers in decimal notation. The pack for Stage (a) uses whole numbers

and tenth-parts. The pack for Stage (b) uses whole numbers, tenth-parts and hun-dredth-parts.

What they do Stage (a) and Stage (b) are played in the same way, except that different packs of number cards are used (see above).

1. The cards are shuffled and put face down. 2. Player A takes the top card, looks at it, but does not let the others see it yet. 3. Player A now points to the appropriate point on the number line, and the others

write the number in place-value notation. 4. Finally the number card used by A is compared with what the others have

written. 5. If these agree, someone else acts as player A. 6. If not they discuss before continuing.

H T Ones t-p h-p

2 7 2 7

2 7

NuSp 1.10 Extrapolation of place-value notation (cont.)

376

activity 4 Shrinking and growing [NuSp 1.10/4]

A group game. This is an Alice-in-Wonderland kind of activity, which will stretch the imaginations of the more able children.

Materials • Number-line 0-10.*† • Card with decimal point, see Figure 47.* • Number cards 1-9.* • A paper clip for a pointer. * Provided in the photomasters † This should be numbered at every whole number, with divisions marked at every

tenth-part and every hundredth-part. This may be made from graph paper with 10 mm and 1 mm squares. The unit interval will then be 1 decimetre, and the number line will be 1 metre long. A small part of this line is illustrated in Figure 47.

What they do Stage (a) 1. The number cards are shuffled and put face down. 2. Player A turns over the top card (say 4) and puts it to the left of the decimal

point. (That is, on top of the zero, so that the zero is not showing but the deci-mal point is).

3. Player B points to the appropriate place on the number line and says “Four ones.”

4. The rest in turn say “Agree” or “Disagree.” 5. Player A moves the card to the right hand side of the decimal point. Player B

should then say “Zero ones, four tenth parts” pointing as she speaks to the ap-propriate point on the number line. The others, as before, say “Agree” or “Disa-gree.’

6. Steps 2-5 are repeated using other number cards. Other players then take over the roles of players A and B, and the activity is repeated.

Stage (b) 1. The number cards are shuffled and put face down as before. 2. Player A now turns over the top two cards and puts one each side of the decimal

point (e.g., 4.7). 3. Player B points to the appropriate place on the number line and says (in this

case) “Four ones, seven tenth-parts.” 4. The rest in turn say “Agree” or “Disagree.” 5. Player A moves both cards to the right hand side of the decimal point, giving

(in this case) 0.47. Player B should then say “Zero ones, four tenth-parts, seven hundredth-parts” pointing as she speaks to the appropriate point on the number line. The others, as before, say “Agree” or “Disagree.”

6. Steps 2-5 are repeated using other number cards. Other players then take over the roles of players A and B, and the activity is repeated.

Stage (c) 1. The number cards are shuffled and put face down as before. 2. Player A now turns over the top three cards and puts one to the left of the deci-

mal point, the other two to the right (e.g., 4.72 ). 3. Player B points to the appropriate place on the number line and says (in this

case) “Four ones, seven tenth-parts, two hundredth-parts.” 4. The rest in turn say “Agree” or “Disagree.”

NuSp 1.10 Extrapolation of place-value notation (cont.)

377

Figure 47 Shrinking and growing

4 5

Number Line

0 .

0 . 4

4 . 7 In use

or

NuSp 1.10 Extrapolation of place-value notation (cont.)

7

378

5. Player A moves all three cards to the right hand side of the decimal point (giving in this case 0.472 ). Player B now has a more difficult task. A suitable response would be to say “Zero ones, four tenth-parts, seven hundredth-parts, two thou-sandth-parts” pointing, as she speaks, to the appropriate point on the number line, and continuing “The thousandth-parts are too small to see.” When I was trying out this activity, Nicholas said “. . . and a very small move to the right.” The others, as before, say “Agree” or “Disagree.”

6. Steps 2-5 are repeated using other number cards. Other players then take over the roles of players A and B, and the activity is repeated.

Stage (d) Activities 1, 2, 3 were about shrinking. Now we come to the growing. They begin

with one number card, as in Stage (a). 1. The number cards are shuffled and put face down. 2. Player A turns over the top card (say 4) and puts it to the left of the decimal

point. (That is, on top of the zero, so that the zero is not showing but the deci-mal point is).

3. Player B points to the appropriate place on the number line and says “Four ones.”

4. The rest in turn say “Agree” or “Disagree.” 5. Player A now moves the card one place to the left, giving (in the present exam-

ple) 40 . This is off the number line, 4 metres from the zero point. So player B might then say something like “Forty. This will be off the line, somewhere in the middle of the next table.” The others, as before, say “Agree” or “Disagree.”

6. Steps 2-5 are repeated using other number cards. Other players then take over the roles of players A and B, and the activity is repeated.

Stages (e), (f) Introduce these at your discretion. The procedure is as in Stage (d), but using two and three cards respectively. The

starting position is always with one card to the left of the decimal point (on top of the zero). Player A then moves them one place at a time to the left, with Player B re-sponding as in step 5 of Stage (d). So in stage (e), she might have to deal with num-bers such as 4.7, 47, and 470; and in stage (f), with the numbers such as 4.72, 47.2, 472, and 4720. In the last case, the corresponding point on the number line would be 4720 decimetres away, i.e., about half a kilometre away. The foregoing would be rather good answers. “Quite a long way down the street” would show sufficient appreciation of the kind of distance involved. This is less complicated to do than to describe, but it is still quite a sophisticated activity, suitable only for older children.

NuSp 1.10 Extrapolation of place-value notation (cont.)

379

Discussion of activities

NuSp 1.10 Extrapolation of place-value notation (cont.)

Shrinking and growing [NuSp 1.10/4]

Here again, children are extrapolating a known pattern: in this case a notation. Since a pattern is shown more strongly by a larger number of examples, Activity 1 shows hundreds and tens as well as ones, and also hundredth-parts although these are harder to point to on the number line. This is also an argument for using mixed numbers. Hundreds, tens, ones firmly establish the tens pattern which is now being extended downwards. Activity 2 is for consolidating the three-way relation between headed column notation, place-value notation, and the decimal fraction concept as it is represented on the number line. The more the connections within a schema (conceptual struc-ture), the better the understanding. In Activity 3, we have gone over entirely to place-value notation. Since the no-tation itself is not new to the children, but only its present extension, we have not spent so long on the way as when it was first introduced. But it is easy enough for you to keep them using headed columns longer if in your judgement they still need the support of this much more explicit notation. Activity 4 is partly for consolidation, but especially for giving children a feeling of the great differences in the sizes represented in this very concise notation. Us-ing a decimetre as unit interval, a digit in the hundreds column represents a length possibly too big to have in the classroom, while the same digit in the hundredth-part column represents a length too small even to draw. This activity also makes frequent demands for adaptability in children’s thinking, so it is a good mental exercise for their intelligence. Is all this talk about “Four ones, seven tenth-parts” etc. really necessary? In my belief, emphatically yes. This verbalization is here being used, in conjunction with the number line, to help establish the concepts. The frequently-encountered problems with ‘decimals’ (decimal fractions in place-value notation) are concep-tual problems, problems of understanding. The importance of these early stages in establishing the right concepts can hardly be over-stated.

OBSERVE AND LISTEN REFLECT DISCUSS

380

Different objects, same pattern [Patt 1.4/1]

381

[Patt 1] Patterns

Patt 1.4 TRANSLATING PATTERNS INTO OTHER EMBODIMENTS

Concepts (i) That the same pattern can be shown by different objects. (ii) That a pattern is therefore independent of any particular embodiment.

Ability To show a given pattern with different objects.

One of the reasons that mathematics is so useful and adaptable a mental tool is the great variety of particular situations to which the same mathematical knowledge can be applied. This means that we must help children’s mathematical concepts to become context-free: that is, independent of any particular embodiment, and, in particular, independent of the manipulatives which are such a useful help in form-ing them initially. The present topic is a good example.

activity 1 Different objects, same pattern [Patt 1.4/1] A teacher-led discussion for a small group. Its purpose is to help children to realize

that the same pattern can be made with a variety of different sets of objects.

Materials • Containers with a variety of objects convenient for making patterns. For ex-ample, one container might have counters in three colours; another might have sunflower seeds, macaroni shells, and dried beans; another might have three different kinds of buttons. There needs to be enough of each variety so that children do not run out while making their patterns: I suggest not less that ten of each. This would allow, e.g., three repetitions of a pattern like

A B B B A B B B . . . with one to spare. [See: ‘Patterns with a variety of objects,’ Patt 1.1/2, in Volume 1.]

Suggested 1. Make a simple pattern using just two kinds of objects from one of the containers. sequence For example, for the shell bean bean shell bean bean shell bean bean . . . discussion 2. Ask if any one thinks they can copy this pattern using objects from one of the

other containers. 3. If no one offers, it may be that they need more time with ‘Patterns with a variety

of objects’ [Patt 1.1/2, Volume 1]. Alternatively, you could make a copy your-self, and ask whether they think that this is the same pattern, although the objects are different. If they appear doubtful, then they do need more time with Patt 1.1/2.

4. Assuming that in step 2 someone does offer, let him try, and ask whether the others agree with what he has made. If they do not agree, discuss what changes are needed so that the second pattern is the same as the first, albeit with different objects.

5. Working in pairs, the children may now choose other containers and make fur-ther embodiments of the given pattern. They check each other’s results.

6. The objects are replaced in the containers, and steps 1 and 5 are repeated.

Discussion of concept

382

activity 2 Patterns which match [Patt 1.4/2] An activity for a small group of children. Its purpose is to consolidate the concept

formed in Activity 1, by applying it to different materials.

Materials • Example patterns from ‘Making patterns on paper,’ Patt 1.1/3 in SAIL Volume 1:

• If available, additional patterns on paper made in Patt 1.1/3 could be included. If not available, and if time allows, have the children make additional repeating patterns following the models in the example patterns (above) .

What they do 1. Begin by showing the example patterns. Ask if any of these show the same pat-tern with different objects.

2. In the set of examples given above, Patterns 2 and 4 show the same pattern with different objects. In short, we say that they match. In this set there are no other matching patterns.

3. The match will show even more clearly if we make a pattern matching Pattern 2 using the letters A B C . . . . If we now do the same with Pattern 4, we find that we get A B C A B C A B C for both.

4. This is also a good way to make sure that we have not overlooked any other pat-terns which are alike.

5. [Optional] Their own patterns (if available) are now spread out on the table. Each child takes a pattern, and collects all the patterns which match it. It is advisable for them to begin by checking that no two of them have matching pat-terns.

6. Finally they check each other’s collections.

activity 3 Patterns in sound [Patt 1.4/3] A continuation of the two previous activities, for a small group. Its purpose is to

show how the concept of pattern can be embodied in a completely different modality.

Materials • Three or more objects which make different sounds when struck. Instruments from a percussion band offer one possibility.

• One pattern from each of the sets of matching patterns from Activity 2.

Patt 1.4 Translating patterns into other embodiments (cont.)

Pattern 2

Pattern 3

Pattern 4

Pattern 5

Pattern 2

Pattern 3

Pattern 4

Pattern 5

Pattern 2

Pattern 3

Pattern 4

Pattern 5

383

What they do 1. The patterns are spread out where all can see them. It would be advisable to begin

with, say, three or four, in which not more than three different objects are used. 2. One child then chooses a pattern without telling the others which it is. He trans-

lates this into a sound pattern, in which each object in the visual pattern corre-sponds to a different sound.

3. The other children try to identify which of the visual patterns is being played. 4. The first child who thinks he knows says so (without indicating which pattern it

is), and demonstrates by continuing the sound pattern. 5. If the second child is right, that child continues until another child thinks he knows

the pattern. If he is wrong, the first child says so and takes back the role of striker. 6. Steps 3, 4, and 5 are repeated until all have identified the pattern. 7. Steps 2 to 6 may then be repeated with a different child starting.

activity 4 similarities and differences between patterns [Patt 1.4/4] A teacher-led discussion for a small group of children. Its purpose is to have chil-

dren reflect on and organize what they now know about patterns, within the frame-work of similarities and differences.

Materials • The example patterns from Activity 2.

Suggested 1. Begin with the example patterns. Ask in what ways they can be alike, and in outline what ways different. for the 2. We have already seen that they can be alike in that the pattern is the same discussion although the objects themselves are different. So we can have two patterns

which use the same objects but are different patterns, or which use different objects but are the same pattern.

3. In the second case, we need a way of saying what the pattern is. I suggest that the translation into letters introduced in Patt 1.4/2 is convenient and universal.

4. Another way in which patterns can be alike or different is the number of differ-ent objects used, and the number of objects in each repetition. These are independent.

For example, A B A A B A A A B A B A A B A A A B A B A A B A A A B uses only two objects, but there are nine objects in each repetition (which we

may call a cycle). The pattern A B C D A B C D A B C D A B C D A B C D A B C D A B C D uses four objects, and the number in a cycle is also four. 5. Given a number of patterns on paper or in sound (though not limited to these)

we can identify the following attributes: The pattern itself. The objects used to show the pattern. The number of different objects used. The number of objects in a cycle. 6. These are also ways in which they can be alike or different, which prepares the

way nicely for the next activity. I suggest that you make a start with this right away, after which they can do it on their own when it is convenient.

Patt 1.4 Translating patterns into other embodiments (cont.)

384

Discussion of activities

activity 5 alike because . . . and different because . . . [Patt 1.4/5] An activity for a small group of children. Its purpose is to consolidate the concepts

which were discussed in Activity 4. Materials • The Activity 2 example patterns, at least, supplemented by others made by the

children, if possible. What they do 1. The patterns are put in a pile face up. The top pattern is then put separately to

start a second pile. 2. The children then take turns describing the likenesses and differences between

the two patterns showing. E.g., if these were the example Patterns 4 and 5, he might say:

“These are alike because they use the same objects: triangles, circles, and squares. They are different because one has a cycle of three and the other has a cycle of five, so they can’t be the same pattern.”

Or if they were Patterns 2 and 4, he might say: “These patterns use different objects, but they both have three different

objects and they both have a cycle of three. They are both the same pattern, which in letters would be A B C A B C A B C . . . .”

The other children say whether they agree or not, and may contribute further suggestions.

3. When the first child has done this, the next child moves the top pattern from pile 1 across to the top of pile 2, so that there is a different pair showing (though one was in the previous pair). He then describes the likenesses and differences between these, as described in step 2.

4. They continue as in step 3 until each child has had at least one turn.

In this topic children are expanding their concept of a pattern using all three modes of schema construction. They are learning from physical experience, using a variety of manipulatives; they are extrapolating pattern concepts to new embodiments; and they are checking for consistency with their existing schemas by frequent discussion.

Patt 1.4 Translating patterns into other embodiments (cont.)

Alike because . . . and different because . . . [Patt 1.4/5]

385

[Meas 1] Length the measurement of distances

Meas 1.4 INTERNATIONAL UNITS: METRE, CENTIMETRE

Concepts (i) The need for international units. (ii) Metre, centimetre.

Abilities (i) To measure in metres and centimetres. (ii) To put this to practical use.

We now introduce the metric system of units, based on the metre. The scientific name for these is S.I. units, where S.I. stands for ‘Système Internationale.’ S.I. units could also be thought of as referring to Standard International units, which would incorporate explicitly the idea of a standard unit. The S.I. system has at least three important advantages. First, it is used internationally by scientists, and by an increasing number of countries for technology, commerce, and everyday life. Second, the system uses base 10 for relating units of the same kind. E.g., one kilometre is 1000 metres, one metre is 100 centimetres; whereas a mile was 1760 yards, a yard was 3 feet, a foot was 12 inches. So S.I. units are much more easily convertible, which is a great advantage for calculations using place-value notation. Third, S.I. units of dif-ferent kinds often have simple relationships. E.g., one gram is the weight of one cubic centimetre of water (at 4 ˚C), one kilogram is the weight of one litre of water (1000 cubic centimetres). We shall gradually introduce children to all of these advantages. Anyone who wants to use standard units needs to have something which is either a copy of the international standard, or a copy of a copy of a copy of . . . this. This is what we get when we buy a metre rule, or a smaller measure. So the use of standard units implies the use of standardized measuring instruments. The accu-racy of these also depends on the transitive property of measurement.

Activity 1 the need for standard units [Meas 1.4/1]

A teacher-led discussion, for any number of children. Its purpose is to show the need for a standard international system of units.

Materials For each table: • A ruler marked and numbered in centimetres.* • A metre-stick. • Source book for historical material. * It does not matter if it is marked in centimetres and millimetres also, but any

numbering must be in centimetres.

Discussion of concepts

386

Suggested 1. Ask: “Who can name one of the countries that we trade with? And another? outline for And another?. . . (Countries of origin of our clothes, watches, playthings, the discussion calculators, . . . ) 2. If we use things like paper clips to measure with, how do we know that they all

use the same size of paper clip as we do? Why does it matter? (Recall Tricky Micky, Meas 1.1/2 in SAIL Volume 1.)

3. What about units for longer distances, such as the length and width of a room, the distance from school to home? Paces would be useful here, but whose? Different people have paces of different lengths.

4. And we need to know how many small units in a larger unit. The number of paper clips in a pace may not be a convenient number. What would be a conven-ient number?

5. So we need at least two units, agreed internationally, one fairly small and one larger. And the large one should be equivalent to a convenient number of the smaller. Convenient for what? Especially, for conversions and calculations.

6. Some will no doubt have heard of centimetres and metres. (Clothes these days are often marked with sizes in both centimetres and inches.) Let them exam-ine a metre rule and a centimetre ruler. Explain that these units are now used in many countries, all over the world.

7. There are 100 centimetres in a metre, just as there are 100 cents in a dollar. In both cases we use the larger unit (metres, dollars) for large amounts, and the smaller units (centimetres, cents) for smaller amounts. And we sometimes use them together, e.g., 3 metres, 45 centimetres; 3 dollars, 45 cents.

8. Canada adopted the S.I. system relatively recently, but the United States has been unable to move as quickly in this direction. Consequently, many of the older units are still encountered (e.g., inches, feet, yards, and miles, as units of distance). Can anyone say how these relate to each other?

9. Within living memory, even more units were in use which, from our present viewpoint, are quite hard to believe. The simplicity and convenience of the S.I. system is brought out more strongly by comparison with these. So I think that it is a worthwhile investigation for children to find out about these and write a short description, with comments.

Activity 2 Counting centimetres with a ruler [Meas 1.4/2] An activity for up to 6 children, to be played in pairs. Its purpose is to establish the

equivalence of counting a number of units, and reading a number on a ruler. This activity should also help children to place the starting point on the ruler correctly.

Materials • Plenty of Centicubes. • A 30 cm ruler for each pair, marked in centimetres.

What they do 1. (Preliminary). All the children make pairs of rods of the same colour and length. No two pairs should be of the same colour. They should vary between 10 cm and 30 cm. These are mixed together and put in the middle for everyone to use.

2. From now on, they work in pairs. Child A has the ruler. 3. Child A names a colour, say blue.

Meas 1.4 International units: metre, centimetre (cont.)

387

4. Each then takes one rod of that colour. ‘A’ finds the number of cubes by count-ing, ‘B’ by using the ruler.

5. Whoever finishes first says the number. The other finishes her way, and says “Agree” or “Disagree.”

6. If “Agree,” they check by putting the rods alongside to see if they match. 7. If “Disagree,” they change roles and measure/count again. 8. If counting cubes and measuring with the ruler do not agree, this might be be-

cause the starting point of the ruler has not been positioned correctly. 9. The rods are then replaced, and steps 2 to 7 are repeated. 10. This activity should be continued until children can always get a correct result

more quickly with the ruler.

Activity 3 Mountain road [Meas 1.4/3] A game for two, four, or six children working in pairs. Its purpose is to use measure-

ments to make predictions of practical interest.

Materials • 3 model trucks, on which loads of different sizes are secured. Some loads are high and narrow, some are wide and low. Furniture vans, trucks with cranes, and the like, could be used for the high narrow ones.

• A table-sized enlargement of the road map on Photomaster 263 in SAIL Volume 2a, with models of bridges replacing the symbols. *

• Centicubes. For each child: • Road map.†

• Ruler marked in centimetres. • Non-permanent marker. * The bridges are made from Centicubes to the heights or widths shown in the

illustration. These measurements should be adjusted to the sizes of the model vehi-cles which you have. The high loads should always be able to cross the bridges of limited width (marked W), and the wide loads should be able to cross bridges with restricted heights (marked H).

† A copy from the photomaster, covered with transparent film to allow use and reuse with non-permanent markers.

Rules of 1. The children are shown the table-sized model of a mountain road. the game 2. Each is given a road map, and the correspondence between this and the road

and bridges explained. The high vehicles can always use the bridges marked W (restricted width), but have to be careful with the bridges of restricted height (marked H). Likewise, the other way around, for trucks with wide loads.

3. Another factor to be taken into account is that the trucks cannot get around hairpin bends.

Meas 1.4 International units: metre, centimetre (cont.)

388

Meas 1.4 International units: metre, centimetre (cont.)

E

D

CB

A

Mountain road [Meas 1.4/3]

So a truck coming from A could go in either direction at junctions C and D, but at B and E it could not take the hairpin turn to the right.

3. Back at their own table, each pair is given a vehicle. One acts as driver, the co-driver acts as navigator.

4. The drivers measure their vehicles, and together they plan their route. They mark this route on their road maps with a non-permanent marker.

5. Watched by the others, each driver in turn takes her vehicle along the chosen route. The co-driver navigates with the help of her road map.

6. Every pair which gets there without mishap, preferably by the shortest way pos-sible for that vehicle, is a winner.

7. The vehicles may be re-allotted, and steps 4 to 6 repeated.

389

Activity 4 Decorating the classroom [Meas 1.4/4] An activity for as many children as you like. Its purpose is to introduce the use of

the metre as a unit for larger measurement.

Materials • Metre measures. * • Party paper streamers. • Plasticine. * These do not need to be graduated in cm, so additional measures can be made from

bamboo, dowel, and the like.

What they do 1. A suitable reason is found for decorating the classroom. 2. In small groups, children are allotted distances in which the streamers are to be

hung. E.g., a wall, the space between two windows. 3. They use the metre rulers to measure these distances, and then to measure the

right lengths of streamer. 4. The streamers may be fixed in position with Plasticine, or otherwise. 5. We are not here concerned with great accuracy, but with the use of a larger unit

for larger distances. So measurements like “3 and a bit over” or “3 and a half” are suitable for this job.

Activity 1 is a general discussion, calling attention to the need for international standard units, and some of the requirements to be satisfied. It relates the knowledge they have acquired while working on this network to their general knowledge. Activity 2 introduces the use of a ruler, which like a number track greatly speeds up the counting of units. It calls attention to the correspondence between counting a number of units and reading this number directly from a ruler, using both Mode 1 (physical matching) and Mode 2 (“Agree”). In Activity 3, I have tried to provide a game which would make use of meas-urement to arrive at a sound plan of action in the same kind of way as in the adult world. This was not easy, since full-sized objects are not easy or convenient for children to move around. Models are much more manageable, and the amount of mathematics used in a given time is greater. But please note that we are not here using scale models. The sizes are actual sizes, in centimetres. If children spon-taneously suggest that 1 centimetre could represent (say) 1 metre, good for them. But conceptually, this is much more advanced, since in their general form scale models use ratio and proportion. At this stage, a centimetre is a centimetre, and the models are objects in their own right. In Activity 4, measurement is put to another of its major uses in the adult world: making things fit. Here we are concerned with introducing the concept, rather than with accuracy. Party streamers provide an convenient and inexpensive material.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 1.4 International units: metre, centimetre (cont.)

Discussion of activities

390

Discussion of concept

Meas 1.5 COMBININg LENgThS CORRESpONDS TO ADDINg NUMBERS OF UNITS

Concept The correspondence stated in the heading.

Ability To make predictions about total length based on sums of individual lengths.

Now that we can use numbers to quantify something continuous, as well as sets of separate objects, we can extend our use of mathematical operations into new realms of application. Indeed, each new kind of unit makes possible a new realm of applications. The first of these applications is in the realm of distance. We find that numbers of units can be added in the same way as pure numbers, and the results put to simi-lar uses. This also works for subtraction. Note that we must be careful when it comes to multiplying. We can multiply a number of units by a pure number. If we put end to end 4 matchboxes each 8 cm long, their total length is 32 cm.

8 cm + 8 cm + 8 cm + 8 cm = 4(8 cm) = 32 cm But (4 cm) * (8 cm) represents an area, as we shall find in Meas 2. This concept could have been understood earlier in the learning sequence. But its application is more powerful when used with standard units, and this is why I have postponed it until here.

Activity 1 Model bridges [Meas 1.5/1]

An activity for up to six children, working singly, or in pairs. This is a continuation of ‘Building a bridge,’ (Meas 1.2/1 in Volume 1). Its purpose is to teach the concept and ability described above.

Materials • River.* • Banks.* • Box girders.* For each child: • A ruler, graduated in centimetres, up to 30 cm. • Boards for the roadway.†

* The river is made of paper or card, painted to look like water, about letter-size (21.5 by 28 cm). The banks can be books, boxes, or the like. The box girders are made from cardboard, square in cross-section. You need 2 each of the following lengths, cut as accurately as possible: 14 cm, 16 cm, 18 cm, 20 cm, 22 cm, 24 cm.

† All the same length, say 15 cm. You need six each of widths 4 cm and 5 cm. The widths should be as accurate as possible.

What they do 1. The task is now to put boards across the box girders, to complete the model bridge. 2. The banks and river are put in the middle, with the plans nearby. 3. Each child has one pair of box girders. He measures the planks, measures the

length of his girders, and works out what combination of planks put side to side will fill the required distance exactly.

391

Discussion of activities

4. He writes this as an addition of numbers of units. This is a good time to intro-duce the abbreviation for centimetre(s), which up till now they have not needed to write. E.g.,

Girders: 18 cm Planks: 5 cm, 5 cm, 4 cm, 4 cm Total: 18 cm 5. Each child in turn then tests his prediction. The banks are adjusted to suit the

length of his girders. He shows the others his calculation, takes the planks as listed, and puts them across the girders to make a roadway. They should fit exactly.

6. Steps 3, 4, and 5 may be repeated, the children using different girders.

Activity 2 how long will the frieze be? [Meas 1.5/2]

An activity for any number of children. It provides a different application of the same concept and ability as Activity 1.

Materials • Newspaper. • Scissors. • Ruler marked in cm. • Metre rule marked in cm.

What they do 1. Each child cuts himself a strip of newspaper, and folds it accordion-wise into 3, 4 or 5 thicknesses. (More, if he likes.) This is trimmed at the free end(s) so that all the folded parts have the same width.

2. He cuts this in any way which leaves part of the folded sides uncut. E.g.:

3. He measures this, and predicts how long his frieze will be when unfolded.

4. He tests his prediction. 5. The total length of several

friezes joined together may also be predicted.

The new concept is a natural extension of their existing concept of addition, and in Activity 1 children are likely to arrive at it intuitively. By putting it to use in a physical embodiment, and testing it by making and testing predictions, this intui-tive knowledge is made more explicit and conscious, and consolidated. Activity 2 is similar. If children multiply rather than add the number of widths to be put together, this is an equally sound method. But in accordance with the discussion of concepts at the beginning of this topic, they should write it in a way which shows correctly what they are doing. I suggest 7 cm * 3 = 21 cm, or 3 ( 7 cm ) = 21 cm.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 1.5 Combining lengths corresponds to adding numbers of units (cont.)

392

Discussion of concepts

Meas 1.6 DIFFERENT SIzED UNITS FOR DIFFERENT jOBS: kILOMETRE, MILLIMETRE, DECIMETRE

Concepts (i) The millimetre as a unit for smaller and more exact measurements. (ii) The kilometre as a unit for larger and less exact measurements. (iii) The decimetre as another useful unit.

Ability To choose and use units suitable for the job to be done.

We need a smaller unit than the centimetre whenever our purpose requires greater accuracy than is given by a centimetre. It also avoids the need for fractional units, for distances less than one centimetre. Likewise, we need a larger unit than the me-tre for the kind of distances we go by car, train, or plane – distances between places, where even the starting and finishing locations are larger than a metre in extent.

For these requirements we use the millimetre and the kilometre, respectively: ab-breviations mm and km. In addition to these, the decimetre provides a convenient unit between the centimetre and the metre.

In this topic the main emphasis is on using the right unit, giving the right degree of accuracy, for the job to be done. The relations between these units will be dealt with more thoroughly in the next topic.

It may be of interest that our word ‘mile’ comes from the corresponding Roman unit of distance, ‘mille passus,’ a thousand paces. In marching, they found it more convenient to count a pace every time the same foot touched the ground, so a Roman pace was what we should call two paces. Their average single pace was shorter than a yard, or we should have had a mile which was 2000 yards, rather than the seem-ingly ridiculous figure of 1760 yards.

Activity 1 “Please be more exact.” (telephone shopping) [Meas 1.6/1]

An activity for children working in pairs. Its purpose is to introduce children to the use of the millimetre as a unit for measuring small distances, where greater accuracy is needed than is provided by the centimetre.

Materials For each pair: • 6 or more nuts and bolts.* • 2 rulers marked (and preferably numbered) in millimetres. * These must all be of different sizes. The differences between their diameters (in-

ner/outer respectively) should be not less than 3 mm.

What they do 1. One player acts as a customer, the other as a hardware clerk. The customer has all the bolts in a bag, the hardware clerk has all the nuts on a tray.

2. The customer takes one bolt out of the bag, and ‘telephones’ the hardware clerk to ask her if she has a nut for it.

3. The customer says “I need a nut to fit a bolt.” 4. The hardware clerk replies “How thick is the bolt?”

393

5. The customer first answers using centimetres. E.g., “A bit less than 2 cm across.” 6. The hardware clerk replies “Please be more exact. Tell me in millimetres.” 7. The customer then measures the bolt and answers, say, “18 mm.” 8. The hardware clerk chooses a nut, checks the inside measurement carefully, and

‘sends’ it to the customer, who checks to see that it fits the bolt. 9. If not, she sends it back. Steps 7 and 8 are repeated until a fit is achieved. 10. This continues until all the bolts are fitted with nuts. Though it is not essential to

include steps 3 to 6 every time, I am in favour of doing so since this little inter-change emphasizes one of the concepts which this activity is intended to teach: the right unit for the job.

Activity 2 Blind picture puzzle [Meas 1.6/2]

An activity for children working singly or in pairs. Its purpose is to consolidate the skill of measurement in millimetres, and the use of measurement to make things fit.

Materials • A shallow box with a transparent plastic panel in the lid. This may also be used to contain the puzzle.

• A number of cut-up pictures no larger than the transparent cutout.* • A fitting guide for each picture.†

* The picture should be cut up into oblong pieces as indicated in the first illustration shown below in ‘Some hints on making . . .’. These widths should differ by amounts of less than 10 mm. It is easiest to cut first, and measure the pieces afterwards. You will need to check that the totals of their widths come to that of the original picture, or at least that the upper and lower rows come to the same total.

† The guide should show only widths and relative locations. The arrows should not be the same lengths as the widths of the corresponding pieces. In other words it is a diagram show-ing sizes, not an accurate pattern. See the SPECIMEN FITTING GUIDE on the next page.

What they do 1. Each child (or pair) has a tray, a lid, and one of the pictures with its fitting guide. 2. The pieces of the picture are spread out face down, and measured one at a time.

Each width is then used, in conjunction with the fitting guide, to put the piece face down in its correct relative position on the transparent plastic lid. This is like doing a jigsaw puzzle ‘blind,’ i.e., without being able to see the picture.

3. The box is put on top, wrong way up inside the lid, so that it will hold the picture in place when the whole is turned over. If all the measurements have been cor-rect, the pieces will be correctly fitted together to make the picture.

The pictures need to be cut into oblongs, in two rows of equal height as shown be-low. It is only the widths which differ, and these are used to decide the locations of the pieces.

Some hints on making the puzzles

Meas 1.6 Different sized units for different jobs: kilometre, millimetre, decimetre (cont.)

394

The locations are shown by a fitting guide like the one shown here. All the widths must be different, by amounts not less than 4 mm.

You may find it convenient to base your own cutting guide on the above. The total width here is 180 mm, so if your picture is (say) 200 mm wide, all you need to do is increase each width by 4 mm. Using the same set of widths, the positions in the upper row can be rearranged to give 120 permutations in all, and likewise for the lower row, making (if you are interested) a total of 14 400 possible sets of dimen-sions for a given overall width.

Interchanging widths in the upper and lower rows is likely to cause confusion, since the overall widths are unlikely to remain the same.

The first puzzle requires a certain amount of trouble to make, but after that it is easy to make more. This activity gives lots of practice in making careful measure-ments, and if these are correctly made and used the result is immediately apparent in the final picture.

Activity 3 “that is too exact.” (Power lines) [Meas 1.6/3]

An activity for two teams. It may also be done by children working in pairs. Its purpose is to expand the concept of ‘the right unit for the job’ in the other direction, that of a larger unit for larger distances. It introduces the kilometre, and also the idea of ‘rounding up.’

Materials • For each team, a diagram of the power lines to be erected. * • Pencil and paper for each child. * Provided in the photomasters

What they do 1. One team forms a party of electrical engineers; the others are in charge of the warehouse and transportation.

2. The engineers decide which line they are going to work on first, and ask the warehouse for a length of cable equal to the distance shown on the diagram. E.g., “Please issue 5 283 metres of cable.”

3. The warehouse reply is “That is too exact. The cable is on reels of 1 kilometre, which is 1000 metres. We’ll give you 6 km of cable.”

4. Here is the warehouse record: Requested Supplied 5 283 m 6 000 m 5. When that cable has been laid, the engineers choose which line is to be built

next, and steps 2, 3, 4, are repeated.

Meas 1.6 Different sized units for different jobs: kilometre, millimetre, decimetre (cont.)

60 mm 10 mm 50 mm 38 mm

56 mm 34 mm16 mm 45 mm

22 mm

29 mm

SPeCIMen FIttIng gUIDe

395

Activity 4 Using the short lengths (Power lines) [Meas 1.6/4]

A continuation of Activity 3, which deals with the obvious question of the lengths left over. It also very simply introduces calculation in these units.

Materials As for Activity 3.

What they do Steps 1, 2, 3 are as in Activity 3. 4. When the partly-used reel is returned to warehouse, a record is made as before

not only of the requested and supplied lengths but also a calculation of how much is left on the reel:

Requested Supplied Surplus 5 283 m 6 000 m 717 m 5. Next time the engineers ask for cable, the warehouse personnel check whether

they have a suitable short length. E.g., in response to a request for 3 578 m of cable, they might reply “We’ll give you 3 one-km reels and a short length of about 717 m.” (“About,” because there may be a small amount wasted.)

6. Steps 1 to 6 are then repeated.

Activity 5 “that is too exact.” (Car rental) [Meas 1.6/5]

An activity for two teams. It may also be done by children working in pairs. Its pur-pose is to use the ideas learned in Activities 3 and 4 in a different situation, and also to introduce the ideas of rounding down.

Materials • Car cards.* * About 10 pictures of cars, stuck on cards. Underneath each is written a test figure

for fuel consumption, e.g., “When tested, this car did exactly 52 837 metres on 1 litre of gasoline.”

What they do 1. One team are customers, the others are car rental agents. 2. The first customer chooses a car, and asks what its fuel consumption is. 3. One of the agents points out that each car has been individually tested, and

points to what is written underneath. 4. The customer replies “That is too exact. My map gives distances in kilome-

tres.” 5. So the agent says (in the example above) “It does 52 kilometres to a litre, and a

bit over.” (Why in this case should he round down?) 6. Steps 2, 3, 4 are repeated with a different customer and agent. 7. The children may also be asked whether they think the cards are sensibly writ-

ten, and suggest possible improvements.

Meas 1.6 Different sized units for different jobs: kilometre, millimetre, decimetre (cont.)

396

Suggested outline for discussion

Activity 6 Chairs in a row [Meas 1.6/6]

An activity for a small group of children working together, say two or three. Its purpose is to introduce the decimetre as a useful unit between the centimetre and the metre. It is introduced by a teacher-led discussion.

1. This needs to be done at a time and place where it will be convenient to move chairs around. The task is to calculate how many chairs of the same size can be put in a row across the room or hall, with at least one metre of free space at each end for access.

2. Discuss what unit should be used. Centimetres would be possible, but are incon-veniently small and unnecessarily exact. Metres are clearly too large, since this is not much different from the width of the chairs themselves. We need some-thing in-between.

3. The decimetre, which is 10 centimetres in length, looks like a good unit for this job, so let us try it out. (How many decimetres are there in a metre?)

What they do 1. The width of a single chair is measured in decimetres, to the nearest unit above. 2. The maximum allowable length is measured, in decimetres. 3. The maximum number of chairs in a row is calculated. 4. This prediction is then tested physically.

You may care to follow this up by discussion of other jobs for which a decimetre would be convenient. For example, the dimensions of wall-to-wall carpeting would need to be measured in centimetres; but for a loose carpet or rug, decimetres would be sufficient.

Activities 1 and 2 both use measurement to make things fit together, which is one of its major uses. The more exact the fit, the more exact are the measurements re-quired. Both of these activities use Mode 1 schema building (physical experience) and also Mode 1 testing (of predictions). Nuts and bolts provide a good example of an exact fit. In some ways this activi-ty is the harder of the two, since it also involves measuring diameters of two kinds, thickness of a bolt and distance across a hole. After some thought, I have put it first because the result of each pair of measurements is immediate and decisive. Activity 2 allows mistakes to persist unchecked until all the pieces are in position. Activities 3, 4, and 5 are somewhat simple. It is difficult to introduce Mode 1 activities involving distances measured in kilometres, so here we rely on Mode 2 building and testing, discussion, and agreement. It should not be difficult to extend the discussion to distances between towns, airports, and other places, in which the locations themselves may be more than a kilometre across. What do we mean, for example, by “the distance from Calgary to Ottawa”? This could have somewhat different meanings according to whether we were talking about a car journey (door to door), or the distance between these cities according to a map. Activity 6 uses the decimetre in a predictive situation.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Meas 1.6 Different sized units for different jobs: kilometre, millimetre, decimetre (cont.)

397

Meas 1.7 SIMpLE CONVERSIONS

Concept That the same distance may be expressed in different units or combinations of units.

Abilities (i) To make simple conversions between units and combinations of units. (ii) To recognize the same measurement written in different ways.

The central idea here is that we do not change the distance itself when we measure it in different units, or when we convert between these. In this topic I have kept to simple conversions, using relationships which the children already know: l km = 1000 m 1 m = 100 cm 1 m = 10 dm 1 cm = 10 mm 1 dm = 10 cm In the next topic we shall see how all of these are related: but for practical purposes, we do not need to use more than two kinds of unit together. (To give a measurement in metres and centimetres implies accuracy of the order of 1 percent; in kilometres and metres, 0.1 percent. So to give a measurement in 3 kinds of units implies a most unlikely accuracy of measurement.)

Activity 1 equivalent measures, cm and mm [Meas 1.7/1]

An activity for children working in pairs. Its purpose is to teach children that the same distance can be measured using different units, and to develop recognition of pairs of equivalent measures. It also gives practice in accurate measurement.

Materials • An assortment of objects to be measured. * • Plasticine (or ‘Blu-tak’). For each child: • Pencil, scrap paper. • A scale marked in cm and mm but not numbered at all.†

* In some objects, e.g., milk straws, length is the salient quality. In others, e.g., ob-longs, ovals, lolly sticks, both length and width should be measured.

† These are easy to make from graph paper. 15 cm is long enough.

What they do 1. The objects are put in the middle for everyone to use. 2. In each pair, one child measures in millimetres only, and the other in centimetres

and millimetres. They begin by agreeing who will do which. 3. Each child chooses an object, and measures its length, or length and width. He

writes his measurement near the bottom of the paper, sticks the object at the top of the paper with Plasticine or Blu-tak, and folds a strip of paper under to hide his writing, like the example at the top of the following page.

Discussion of concept

398

length 6 cm 8 mm width 4 cm 5 mm

first this then this

4. Each then changes with his partner, who measures the object on his paper again, and writes the result in his own way below the object.

5. They then unfold the bottom strip, and compare results.

6. If the results do not match in this way, they show each other how they have measured and discuss.

7. It may need practice before children can measure accurately to a millimetre, and we need to separate mismatches due to this difficulty from those due to mistakes about units. For example, if Paul records 67 mm and Mary records 6 cm 8 mm, you could ask Mary “What would yours need to be, to agree with Paul’s? Not to be all in millimetres, but to agree?” And a similar question could be put to Paul.

8. Steps 3 to 6 are repeated, with other objects.

Activity 2 Buyer, beware [Meas 1.7/2]

A shopping activity for children working in pairs. Its purpose is to consolidate the concepts and abilities learned in Activity 1 in a social situation which does not de-pend on actual measurement.

Materials • Two notices, as illustrated in the photomaster provided in SAIL Volume 2a.

Meas 1.7 Simple conversions (cont.)

object object

length 6 cm 8 mm width 4 cm 5 mm

length 68 mm width 45 mm

object

399

What they do 1. Children form pairs, in which one acts as parent and the other, a little child (about a year younger than the child taking the part of the parent).

2. They are walking along the sea front or fair ground when the little child sees the notice MORE FOR YOUR MONEY, and says: “Mom/Dad, I’ve got some money left. Can I buy a 12¢/14¢/20¢ stick of rock candy? (whichever he or she likes).

3. The parent sees the other notice nearby and says that the stall with the notice ALL PRICES PLAINLY MARKED gives better value. The child acting as a little girl/boy pretends not to understand why this is so, until the parent has ex-plained so clearly that one could hardly fail to ‘catch on.’

4. Two more children take the parts of parent and child, and steps 2 and 3 are re-peated.

5. Children may then make up some misleading notices of their own, with plainly marked prices for comparison.

This activity should now be repeated with another commodity, such as lengths of

wood, the alternative prices now being in centimetres at the overpriced stand, and in decimetres at that of the better-value stand.

Activity 3 A computer-controlled train [Meas 1.7/3]

An activity for children working in pairs. Its purpose is to introduce conversion both ways between distances given in kilometres and metres, and in metres only.

Materials • Card representing computer screen, letter size.* • List of stops and distances.†

• Card giving choice of prepared screen messages.†

• Non-permanent marker. • Wiper. • Pencil and paper for each child. * See illustration below. † Provided in the photomasters.

The card with prepared screen messages is threaded through the slits, showing the chosen message on screen.

The whole may be laminated or covered with a blank sheet of clear acetate.

Meas 1.7 Simple conversions (cont.)

COMPUteR SCReen

SLITS

400

What they do 1. One child acts as a computer-controlled train, the other as the controlling operator.

2. The train is controlled to stop at whatever distance is entered, to the nearest metre. (The nearest kilometre would hardly be accurate enough!)

3. The activity begins with a screen display of the prepared message: WHAT IS DISTANCE TO NEXT STOP? 4. The controller writes the distance on the screen, using the non-permanent

marker. The first time, he writes this as it appears on his list, which is the way humans understand more easily.

5. In this case, the computer responds with the screen message PROGRAM WORKS ONLY IN METRES 6. So the controller has to convert into metres. This is quite easy, since each km is

1000 m. E.g., Distance is 7 km 437 m 7000 m + 437 m Distance is 7437 m 7. The controller writes this on the screen. 8. The computer changes the message to TRAIN MOVING UNDER COMPUTER CONTROL 9. At any time while this is on screen, the controller may change the message to REPORT DISTANCE AFTER LAST STOP 10. The computer clears the screen of the distance(s) previously written, and writes

any number less than the distance it is supposed to go before stopping. This gives distances in metres, which the controller has to change into kilometres and metres.

11. If this distance appears unlikely, the controller puts on screen the message EMERGENCY– ERROR SUSPECTED 12. The computer checks, and puts a corrected distance on screen. 13. When the trains has reached the programmed stop, it first clears the screen of all

numbers and then puts on screen ARRIVED followed by WHAT IS DISTANCE TO NEXT STOP? 14. Steps 3 to 13 are then repeated. (Steps 4 and 5 are optional, but the controller

still has to convert each distance before entering it.) 15. At step 7, the controller may write a number which does not match the

computer’s internal check with expected distances. In that case the emergency message (step 11) comes on screen. This time it is the controller who has to check.

Meas 1.7 Simple conversions (cont.)

401

Activity 1 relates every pair of equivalent measures to the same object, which em-phasizes the central concept of conversion. This is Mode 1 concept building, with Mode 2 testing.

Activity 2 underlines the point that if we look at the number alone, without thinking about the units, we are liable to be misled. This activity is more abstract, as introduced. But I think that the parents who are most successful in explaining will be those who show their children what these lengths actually look like. (They might also point out that the stall-keeper is unlikely to have measured the rock to an accuracy of a millimetre.) In Activity 3, the conversions are again straightforward, provided that care is taken over zeros.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 1.7 Simple conversions (cont.)

Discussion of activities

same distance

measure measure stated one way stated another way

do these agree?

A computer-controlled train [Meas 1.7/3]

402

Meas 1.8 ThE SySTEM OVERALL

Concept An overview of how kilometres, metres, centimetres, and millimetres all relate to each other.

Abilities (i) To convert any distance to metres. (ii) To recognize equivalent measures.

The use of base 10 throughout the standard international system fits very nicely with place value notation, including decimal fractions. So 5 cm 3 mm can be writ-ten 5.3 cm, since a millimetre is a tenth-part of a centimetre. From a practical point of view I see little or no point in doing this, since a major advantage of the system is that we can avoid fractions by using smaller units. Why write a distance as 0.47 m when we can write it more simply as 47 cm? However, it seems a natural conclusion for this network to let children see an overview of how all the units they have learned relate to each other. So the activi-ties of the present topic have been designed with the latter as its main purpose. The only use of decimal fractions will be to exemplify relations of other units to the metre.

Activity 1 Relating different units [Meas 1.8/1]

An investigation for children working singly or in small groups. Its purpose is to help children to an overview of the relationships between all the units they have learned so far.

Materials • Headings card.* • Pencil and paper for each child * Illustrated below, in step 2.

What they do 1. Ask them first to write down everything they know already about kilometres, metres, centimetres, and millimetres. E.g., 1 km = 1000 m. They should check by comparison with each other.

2. Then ask them each to copy this heading from the card.

3. Ask what they think is the main unit in the metric system, on which all the other units are based? (Answer, the metre. The word ‘metric’ tells us.)

4. So they all write the abbreviation m in the ones column. 5. Ask them if they can put km, dm, cm, mm in their correct columns, using what

they wrote down earlier.

Discussion of concept

Th H T Ones t-p h-p th-p

403

6. When finished, they again check by comparison. Their results should look like this:

7. These headed columns should be kept for the next activity. 8. Explain that ‘kilo’ means a thousand times, ‘deci’ means a tenth-part, ‘centi’

means a hundredth-part, and ‘milli’ means a thousandth-part. Also, that al-though there are names for the other spaces, these are not often used.

9. On future occasions, t-p, h-p and th-p may be used as abbreviations for the fractional parts. Note the use of lower case letters for these, and capitals for the multiples.

Activity 2 “Please may I have?” (Metre and related units) [Meas 1.8/2]

A card game for 5 or 6 children. Its purpose is to exercise their thinking about the relations between the metre as main unit and kilometres, centimetres, and millime-tres. It requires previous knowledge of decimal fractions.

Materials • Pack of cards (metre and related units).* • The headed columns from the last activity. * Provided in the photomasters. These are doubled-headed, 30 in number, with the

following inscriptions:

Pack 1 Pack 2 1 km 1000 m 5 km 5000 m 1 m 10 dm 100 cm 1000 mm 5 m 50 dm 500 cm 5000 m 1 dm 10 cm 100 mm 0.1 m 5 dm 50 cm 500 mm 0.5 m 1 cm 10 mm 0.01 m 1 cm 5 cm 50 mm 0.05 m 5 cm 1 mm 0.001 m 5 mm 0.005 m To get a pack of 30, use either two sets of Pack 1 with one pair removed (easier)

or one Pack 1 and one Pack 2 (harder). Players should note that the only fractions which occur are fractions of a metre. The game may be played first with, and then without, the visual aid of a metre-stick.

1. All the cards are dealt. This will give 6 or 5 cards, each, for 5 or 6 players, re-spectively.

2. Players look at their cards. The object is to collect pairs of equivalent measures, e.g., 1 cm and 0.01 m. But identical measures, e.g., 1 cm and 1 cm, are not al-lowed.

3. At the start of play, players put down any pairs which they have. 4. They then ask in turn for a card which would enable them to make a pair.

Th H T Ones t-p h-p th-p km m dm cm mm

Meas 1.8 The system overall (cont.)

404

5. E.g., if Sean holds a card showing 10 cm, he might ask “Please, Tina, may I have zero point one metres?”

6. Tina gives this card to Sean if she has it. Otherwise she says “Sorry.” In that case, the player who does hold 0.1 m might deduce that Sean holds the 10 cm card, thereby ensuring a pair when her turn comes up.

7. When a pair is collected, it is put on the table, face up, for the others to check. The winner is the first to put down all her cards. The others may play out their hands, however.

The last two are not practical activities directly connected with measurement, but activities which involve first thinking about, and then using, the interrelations of the units which they have been using. They are activities in the use of reflective intelligence.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 1.8 The system overall (cont.)

Discussion of activities

Relating different units [Meas 1.8/1]

405

[Meas 2] AreA The measurement of surfaces

Meas 2.1 MEASURING AREA

Concepts (i) Tiling a surface. (ii) Measuring area by counting tiles. (iii) The square centimetre as a standard unit. (iv) The centimetre grid as a measuring device.

Abilities (i) To tile a surface in a variety of ways. (ii) To measure a surface by counting tiles. (iii) To measure a surface by using a grid of centimetre squares.

The term ‘area’ is in general use with two meanings: a region (“There is an area of grass in front of the town hall.”), and the size of a region (“The area of our garden is approximately 600 square metres.”). Usually the context makes it clear which meaning we are using. If we want to distinguish explicitly between the two mean-ings, we can say that area is the measure of a surface, in the same way as length is the measure of a distance. Whatever we choose for our unit of area must be easily reproducible, and we must be able to cover the surface with a number of these, without gaps and without overlap. (If there are gaps, some of the surface is not included. If there is overlap, some is included twice.) Doing this is called tiling, or tessellating. And the simpler the relationship with other units, the better. (Simpler than 2240 square yards = 1 acre!) The square centimetre satisfies these requirements well for small areas.

Activity 1 “Will it, won’t it?” [Meas 2.1/1] A teacher-led activity for a small group of children. Its purpose is to introduce chil-

dren to the first requirement of a unit for area measurement, that it can be used for tiling a surface.

Materials • Eight different sets of identical shapes, of which some will and some will not tile.*

* Suitable sets are provided in the photomasters.

What they do 1. Start with the set of circles. Use these to demonstrate how if we try to cover part of the table top with these, either they leave gaps, or they overlap.

2. Put these away and give the children one set each. Begin with the oblongs. Ask the child who has these to put just one of them in the middle. The children then say in turn whether they think that these shapes will or won’t fit together exactly, without gaps and without overlap. When all the children have replied, the child with the oblongs tests their predictions using the rest of the set.

Discussion of concepts

406

3. Explain that when (as with the oblongs) we can cover a surface with these with-out gaps or overlap, this is called tiling.

4. In turn each child similarly puts out one shape from his set, and when all have said whether or not it can be used for tiling a surface, tests their predictions us-ing the rest of the set. When it is clear that they have the idea, they may be left to continue on their own.

5. Note that the triangles must all be the same way up. The markings are there primarily for this purpose, but some children may make other interesting discov-eries.

Activity 2 Measuring by counting tiles [Meas 2.1/2]

A teacher-led discussion for a small group. This is to introduce the idea of measur-ing the area of a surface by counting how many tiles are required to cover it. It fol-lows immediately on from Activity 1, and need not take long.

Materials • The tileable shapes from Activity 1 should be available for use if needed. • A sheet of blank paper for teacher. 1. Draw an irregular shape, smaller than 10 cm by 15 cm (anticipating Activity 4),

and ask how we could use tiling to measure how big it is. 2. The answer is, of course, to tile it and count how many tiles it takes. 3. There are however two questions which arise. One is that different tiles will

give different answers, so we shall have to choose one as a standard which every one uses. This is discussed in the next activity. The other is that the tiles will not fit tidily at the edges of the region. This will be dealt with in Activity 4.

Activity 3 Advantages and disadvantages [Meas 2.1/3] A teacher-led discussion for a small group. Its purpose is to introduce the square

centimetre as the standard unit for small areas.

Materials • As for Activity 2, and also • Some 1 centimetre square tiles.

1. Each child uses one shape to give an example of tiling with that shape. 2. They discuss their advantages and disadvantages. For example, the hexagons

are not easy to make lots of. The squares and oblongs are easy to use, but are not yet of any standard size.

3. The 1 centimetre squares fit in exactly with the unit already in use for length. They are however very small and fiddly to use. Apart from this disadvantage, it is a natural choice of unit.

Meas 2.1 Measuring area (cont.)

Suggested outline for the discussion

Suggested outline for the discussion

407

Activity 4 Instant tiling [Meas 2.1/4] A demonstration and teacher-led discussion for a small group. It introduces children

to the use of a centimetre grid as an instrument for measuring area.

Materials • The shape you drew in Activity 2, step 1 (or another if you prefer). • A centimetre grid on transparent material. * • Non-permanent marker. * A photomaster is provided in SAIL Volume 2a. If you have access to a suitable

copier, the grid can be copied on to a transparency.

1. Show the shape whose area is to be measured. As they already know, this is to be done by counting the number of unit tiles, each 1 cm square, which will cover it.

2. Produce the centimetre grid, and by putting this on top of the shape show how it instantly covers it with unit tiles. All we now have to do is count them.

3. There is still the problem of the part-squares around the boundary. These can be dealt with in two ways. One is to count those which are more than half as wholes, and not to count at all those less than half. The other, which is more accurate, is mentally to combine part-squares to make whole squares.

4. Whichever method is used, one needs a washable marker to keep track of which squares and part-squares have been counted.

In this topic we aim to establish the true concept of area, as the number of unit squares which can be combined to cover the given surface. Area is not length multiplied by width, as so many children incorrectly learn. This is just a con-venient shortcut for finding this number of unit squares, which applies only to rectangles. The first three concepts listed are closely related, and so are the activities. The last activity provides a convenient technique for covering the given region with unit squares. All three make full use of Mode 1 schema building from activities with physical materials, and testing of predictions; and of Mode 2 schema con-struction, by communication from a teacher and discussion.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Meas 2.1 Measuring area (cont.)

Suggested outline for the discussion

408

Meas 2.2 IRREGULAR SHAPES WHICH DO NOT FIT THE GRID

Concept Consolidation and expansion of the concepts learned in topic 1 to new situations.

Ability To measure the area of a surface with an irregular outline, by counting unit squares.

No new concept is introduced in this topic. The aim is to consolidate the concept of area in its general form, before teaching short-cuts for particular cases.

Activity 1 Shapes and sizes [Meas 2.2/1] An activity for a small group of children. Its purpose is to give them practice in the

use of the centimetre grid for measuring area.

Materials • Six cards with outlines of cookies of different shapes.* • Centimetre grid transparency. • Non-permanent marker and wiper for each child. * A suitable set is provided in the photomasters.

What they do 1. Each child receives one card at random. 2. The first child puts her card where all can see it. 3. The next child looks at her own card, and tries to decide whether it is smaller

or larger in area than the one already there. If smaller, she puts it in the left; if larger, on the right.

4. The other children in turn put down their cards at the end of the row, or between others, in what they think is the order of size.

5. The order is recorded, and each child then takes one card and measures its area by using a centimetre grid using the method described in Meas 2.1/4.

6. Finally the cards are arranged in the correct order of size. Any differences be-tween this and the order recorded in step 5 may then be discussed.

7. If desired, steps 1 to 6 may be repeated.

Activity 2 “Hard to know until we measure” [Meas 2.2/2] An activity for two small teams of children. Its purpose is to give them further prac-

tice in the use of the centimetre grid for measuring area, and in estimating relative areas.

Materials • Sixteen cards, each providing part of an outline for a surface, as illustrated.* • Non-permanent marker and wiper for each team. • Pencil and paper. • Centimetre grid transparency. * A suitable set is provided in the photomasters.

Discussion of concepts

409

What they do 1. The cards are spread out where all can see them. The first team then chooses four, and puts them together to enclose an area, as shown in the illustration.

2. The second team then uses four of the other cards to make another outline, to enclose an area as nearly as possible equal to the other.

3. Are they of the same area, or is one larger and, if so, which? It is hard to tell until they measure.

4. So each team measures the shape they made, and the results are compared. 5. Steps 1 to 4 may then be repeated.

Activity 3 Gold rush [Meas 2.2/3] (This embodiment was suggested by Helen Barrett, teacher at the International

School, Hamburg.)

Part (a): A harder activity, for children to work at individually. It takes the form of an investigation, gives much practice in measuring area by counting squares, and thereby further consolidates the concept of area as already described. Plenty of time is likely to be needed.

Part (b): A teacher-led discussion, when all have completed Part (a).

Part (a) Materials For each child: • A loop of thin string,* pipe cleaner, or waxed string (e.g.,Wikki Stix). • A centimetre grid. • Non-permanent marker and wiper. • Pencil and paper. * Cotton is better than plastic, which is too springy. String lies better if wetted.

What they do 1. They are asked to imagine that they are prospectors in the gold rush. Claims may be of any shape, but the maximum length of boundary allowed is fixed. This is represented by their loops of string. (You could also tell them that the word for distance all the way round is ‘perimeter.’)

2. Their first task is therefore to find out what shape they should make their claims in order to enclose the largest area of ground. For this they use their loops of string and centimetre grids in the way they have already learned.

3. They work individually, and keep their results secret from their rival prospec-tors. When they think that they have found the best shape, they tell their teacher without letting anyone else know.

Meas 2.2 Irregular shapes which do not fit the grid (cont.)

410

Part (b) Suggested 1. Point out that what they have found out will be much less useful if the best shape outline for varies according to the length of the boundary. To find out if it does, the lengths the discussion of their loops are not all the same. 2. We assume that they have all arrived at the correct result, a circle. This answers

the question for their loops of string. Is the answer still true for larger loops? 3. What if we used an overhead projector, to show their loops much bigger on a

screen? Would the result still be the same? 4. And can they imagine a giant overhead projector, to show their loops on the

ground? Would the result still be the same? 5. The expected answer in both cases is “Yes.” Though the foregoing is not a

rigorous proof, it is good mathematical intuition, and can be made into a logical mathematical proof based on similar figures.

The first two activities involve Mode 1 learning, in this case consolidation of the concept by practical activity, and Mode 2 testing, discussion within the peer group. The third activity brings in Mode 3 building (creativity), since they have to come up with new possibilities in a systematic way. Mode 2 building and testing then follows in the teacher-led discussion.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 2.2 Irregular shapes which do not fit the grid (cont.)

Discussion of activities

Gold rush [Meas 2.2/3]

411

Meas 2.3 RECTANGLES (WHOLE NUMBER DIMENSIONS); MEASUREMENT BY CALCULATION

Concept That in certain cases, the number of unit squares can be found by calculation.

Ability To calculate the area of a given rectangle.

The general concept is as stated above. The method of calculation for a rectangle is of course well known at an instrumental level, but after topics 1 and 2 we are now in a position to understand it relationally, and to distinguish between this particular method and the concept of area itself. ‘Length times breadth’ does however provide us with an excellent example of the lack of adaptability of instrumental understanding, since this pseudo-concept of area is of no use for any other shapes.

Activity 1 “I know a shortcut.” [Meas 2.3/1] A teacher-led activity for a small group of children. Its purpose is to introduce chil-

dren to the above concept for the particular case of rectangles.

Materials • About six different sets of identical cutout rectangles whose dimensions are a whole number of centimetres. In each set there needs to be one per child and one for the teacher.*

• A centimetre grid for each person.* * Provided in the photomasters

What they do 1. Each person has one copy of the same shape. The aim is to be the first to say its area.

2. Before starting, they are told that for rectangles there is a shortcut by which they can be fairly sure to come first if the others do not know it. If they think they know it, they shouldn’t say what it is.

3. When all have given their answers, they re-check if these do not agree. Then, if the first to answer was correct, he drops out of the race provided he did so by using a shortcut. (The first time this will, of course, be yourself.) He does not however drop out of checking.

4. The last is given the choice of being told the shortcut, with explanation; or of continuing to try.

5. Finally they check whether they were using the same shortcut. I only know of one, so this might be interesting!

Discussion of concepts

412

Activity 2 Claim and explain [Meas 2.3/2]

An activity for a small group of children. Its purpose is to give practice in using the method with understanding.

Materials • A set of cards with rectangles of various sizes all marked in centimetre squares. These are in pairs of equal area, e.g., 4 cm by 4 cm and 8 cm by 2 cm.*

* A suitable set is provided in the photomasters.

What they do 1. The pack is shuffled and put face down on the table. The top card is turned over and put centrally.

2. In turn, the players turn over the top card of the pile and put it centrally. If they see a pair of the same area, they claim it and explain how they know that the area is the same. (“Eight rows, two in each, sixteen square centimetres” is a better explanation than, say, “Eight twos are sixteen.”) If they cannot see a pair, they say “No claim.”

3. The player whose turn it is may begin by claiming any pair which has been over-looked. He then still has his turn as in step 2.

4. In the event that there is no face-up card when it comes to a player’s turn, he may turn over two cards.

5. When all the cards have been turned over, the winner is the player with most pairs.

Activity 3 Carpeting with remnants [Meas 2.3/3]

An activity for a small group of children. Its purpose is to give further practice in finding the area of rectangles by calculation.

Materials • For each player, a rectangle drawn on paper which is ruled in cm squares. The sides of these rectangles should fit the lines on the paper.*

• Two lots of assorted rectangular cards, one larger and one smaller. These are not ruled in squares.**

• Pencil. * The sizes are not critical, but, as a guide, a suitable set to start with is included in

the photomasters, measuring: 7 cm x 16 cm, 11 cm x 16 cm, 10 cm x 11 cm, 7 cm x 21 cm, 11 cm x 11 cm, 8 cm x 15 cm, and 11 cm x 14 cm. It would fit the narrative if the children were given the sizes and drew their own rectangles.

** Suitable examples are provided in the photomasters. What they do 1. Each player’s rectangle represents at one-tenth size a floor to be carpeted as

cheaply as possible by using remnants. These are represented by the smaller cards, which are roughly sorted into smaller and larger for shoppers’ conven-ience. These are likewise one-tenth size.

2. Each player in turn chooses a remnant, and shades in a rectangle of the same area in his diagram. This need not be the same shape, but it must be a single rectangle. Once chosen, a remnant may not be returned.

3. The last rectangle need not provide exactly the right area, but players must record the amount left over since the aim is to minimize waste.

Variation They still choose in turn, but do not shade their rectangles until finished. In this

version, they may then return one remnant in exchange for one other, once only.

Meas 2.3 Rectangles (whole number dimensions); measurement by calculation (cont.)

413

In all these activities, the relation between the area as found by calculation and the area as a number of unit squares is preserved. Activity 3 introduces the idea of a scale drawing at an intuitive level, but this is not made explicit. Children who know about them will notice that the markings on the materials represent decime-tres (which in my view are very useful units).

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 2.3 Rectangles (whole number dimensions); measurement by calculation (cont.)

Discussion of activities

Carpeting with remnants [Meas 2.3/3]

414

Meas 2.4 OTHER SHAPES MADE UP OF RECTANGLES

Concept No new concept, but expansion to more difficult examples. Abilities To calculate areas which can be seen as made up of rectangles, either by adding or

subtracting.

Having started with the simplest case of a surface whose area can be found by calculation from linear measurements, namely a rectangle, we start on the road of expanding the range of examples to which this can be applied.

Activity 1 ‘Home improvement’ in a doll’s house [Meas 2.4/1]

An activity for children to work on individually. Its purpose is to help children to enlarge their range of application of the method learned in the previous topic. It is introduced by a teacher-led discussion.

Materials • Dimensioned floor plans of several rooms in a doll’s house (not drawn to scale). It is useful to have two or three copies of the same plan. These may conven-iently be laminated on card.*

• Pencil and paper for each child. * Photomasters are provided for a kitchen, a bathroom, and a utility room.

What they do 1. Begin by showing them one of the plans, explaining that it represents the floor of a room in a doll’s house. This needs to be covered with tiles 1 cm square. We want to know how many are needed, so that the right number can be bought.

2. We cannot use a centimetre grid, since the diagrams are not full size, nor are they to scale as they were in Meas 2.3/3, Carpeting with remnants.

3. There are two general approaches. One is to split up the area to be measured into rectangles, calculate the areas of each of these separately, and add. The other is to start with (in this case) the area of the whole floor, and then subtract all the areas not to be included. Usually one of these is neater than the other, the subtraction method being the less obvious.

4. When they have the idea, give out plans for them to work on individually. Those with plans for the same room can then compare their results, and discuss each other’s methods.

Discussion of concepts

415

Activity 2 Claim and explain (harder) [Meas 2.4/2]

A harder version of Meas 2.3/2. Its purpose is to give further practice in the method introduced in Activity 1.

Materials • A set of cards like those used in Meas 2.3/2, except that one card of each pair shows not a single rectangle but a shape made up of rectangles, e.g., a frame, border, L-shape. Some of these should be easily rearrangeable into a rectangle, thus preparing the way for the next topic.*

• They will probably find pencil and paper useful sometimes. * A suitable set is provided in the photomasters. What they do This game is played in exactly the same way as the earlier version (Meas 2.3/2),

except for the different set of cards.

Note Some of these cards show a blank area surrounded by a frame, such as might rep-resent a picture frame, a tiled pool deck, or a path around a rectangular flower bed. These are well-known examples for which the subtraction method is much quicker and neater. At school we were taught to find “the whole minus the hole”! I wonder if this is still in use, after more than 60 years?

Activity 3 Net of a box [Meas 2.4/3]

An investigation for children working in small groups. Its purpose is for children to apply the method learned in Activity 1 in a new situation, and to give them further practice.

Materials • Two empty boxes (e.g., cereal boxes), with the tops stuck down again. One is

left like this, the other is cut along some of its edges so that it can be spread out flat to give what is called a net. This can be done in several ways, and any one of these will do.

• A dimensioned drawing of the net, large enough for all the children to see, with the dimensions rounded to the nearest centimetre.*

* A ready-made, dimensioned net for a box is provided in the photomasters.

Introduction 1. Show them the box and its net, and tell them the term for the latter. 2. Explain that the net is to be cut out of a single rectangle of card, the smallest

possible.

The investigation First part In the example that you have given them, what is the area of card used, and what is

the area wasted? Second part Can the waste be reduced by using a different net of the same box? Third part Some children may like to investigate whether, if a lot of boxes are to be made,

waste can be reduced by tessellating a number of these nets on a large sheet.

Note If anyone points out that a box cannot in fact be constructed from a net of this kind, they are of course quite right. We may then explain that we need to begin with a simplified example. The same principle will apply if we include all the tabs for sticking, etc. (except perhaps in the third stage of the investigation).

Meas 2.4 Other shapes made up of rectangles (cont.)

416

Activity 4 Tiling the floors in a home [Meas 2.4/4]

A more difficult version of Activity 1, for the more able children to work on individ-ually or in pairs. It also provides an opportunity to introduce the square decimetre. Although this unit does not seem to be widely used, the (linear) decimetre is intro-duced in Meas 1 (topic 6), so it would seem sensible at least to mention the square decimetre. The activity is introduced by a teacher-led discussion.

Materials • Dimensioned floor plans of three floors in a house (not drawn to scale). It is useful to have two or three copies of the same plan. These may conveniently be laminated on card.*

• Pencil and paper for each child. * Provided in the photomasters

What they do 1. Begin by showing them the plans, explaining that each represents one floor of a house. The exposed parts need to be covered with tiles, and we want to know how many of these are needed.

2. The tiles are available in different materials: carpet, vinyl, and non-slip ceramic. Suggestions may be invited as to which would be suitable for different rooms.

3. The tiles are all 10 cm square, and (very conveniently) the dimensions of the surfaces to be covered are all multiples of 10 cm.

4. The floor plans are now given out for children to work on individually or in pairs. Those with plans for the same room then compare their results, and dis-cuss each other’s methods.

5. If this has not already arisen spontaneously, ask what we would call the area of one tile. So the floor surfaces to be covered measure . . .? What is this in square metres?

These activities evoke children’s creativity in extending the range of examples for which a desired area can be calculated by using the simple method for calculating the area of a rectangle. For Activity 2, mental agility is also needed. Since these are consolidation activities, I suggest that children are given plenty of practice in them, so that they may become fluent in adapting the method to many particular instances.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 2.4 Other shapes made up of rectangles (cont.)

Discussion of activities

417

Meas 2.5 OTHER SHAPES CONVERTIBLE TO RECTANGLES

Concepts (i) That other shapes can be converted to rectangles of the same area. (ii) That in this way, their areas can also be found by calculation.

Ability To calculate the areas of given parallelograms, triangles, and circles.

Here we continue along the road of calculating areas of surfaces from linear meas-urements. The beginning made in topic 3 is here expanded further, first to two more shapes whose sides are straight lines, and then to circles. To convert a circle into a rectangle of the same area seems to me an attractive example of mathemati-cal ingenuity, and it also introduces the concept of ‘getting closer and closer to . . .’ which is mathematically quite advanced.

Activity 1 Area of a parallelogram [Meas 2.5/1]

An activity for children working in small groups. Its purpose is to help them find and use a method for calculating the area of a parallelogram.

Materials • For each child, a sheet of coloured paper on which several parallelograms are drawn. In each parallelogram, the lengths of one pair of opposite sides and the perpendicular distance between them are whole numbers of centimetres.*

• For each child, a straight-edge, a sheet of plain paper, and a pencil. • For the group, several pairs of scissors. • For the group, several gluesticks or equivalent. * A suitable set is provided in the photomasters.

What they do 1. Each child cuts out the top parallelogram on their sheet (A). 2. Tell them that by making just one cut and rearranging the pieces, their parallelo-

gram can be converted into a rectangle of the same area. 3. They all have a try, and then share their ideas and results. The successful con-

versions are pasted at the top of their sheets of paper. When they discover the importance of an accurate right-angle, they may find the corner of a sheet of paper helpful.

4. By now we hope that at least one of the group has found the well-known method illustrated below, and thereby the area of the parallelogram.

Discussion of concepts

418

5. Ask them to find the area of the next parallelogram (B on the Meas 2.5/1 photo-master) by the same method, and compare results.

6. There are four more parallelograms whose areas are to be found. Tell them that there is a quicker way which doesn’t need cutting. Ask them to think about it, and then to share their ideas. When a good method has been agreed on, they use it to find the areas of parallelograms C and D, and compare results.

7. There are two stages of this shortcut. The first is to replace cutting and pasting by drawing. The second is to by-pass this and simply calculate the area of the equivalent rectangle, without actually drawing it. If someone goes straight to the second stage, well done, but he should be asked to explain how he got there. This end result is easier to reach if the cut has been positioned as in the left-hand pair of diagrams.

8. Finally they use their new method to calculate the areas of the two remaining parallelograms, E and F.

Notes 1. For many parallelograms there are two directions in which the cut can be made, as shown here. Both are valid. In the examples provided in the photomasters, only one of these gives whole-number measurements.

2. We need a summary of the method which has been arrived at. The conventional one is ‘base multiplied by height,’ which is convenient provided we remember that the base can be any side, and the height has to be perpendicular to it and not straight up and down the page. As an alternative I suggest ‘The length of a side multiplied by the perpendicular distance from its opposite side.’ More accurate, but cumbersome. The first might be accepted provided that, if challenged, the user can expand it into something like the second.

Meas 2.5 Other shapes convertible to rectangles (cont.)

419

Meas 2.5 Other shapes convertible to rectangles (cont.)

Activity 2 Area of a triangle [Meas 2.5/2]

An activity for children working in small groups. Its purpose is to help them find and use a method for calculating the area of a triangle.

Materials For the group: • One or more pages of identical right-angled triangles, suitable in size for cutting

out if desired. There should be enough of these for each child to have two if they want.*

• One or more pages of scalene triangles.* • Several pairs of scissors. • Several gluesticks or equivalent. For each child: • A sheet of plain paper, and a pencil. * A suitable set is provided in the photomasters.

What they do Stage (a) Right angled triangles 1. Someone cuts up enough of the sheet(s) of right angled triangles for each child

to receive an oblong of paper with a triangle on it. There should be some spare ones also. These are placed centrally.

2. Each child takes one of these, cuts it out, and tries to find its area by whatever means they like. Tell them that they may take another triangle if they wish and that the easiest way does not involve cutting up the triangle. The other materials are there for use if required.

3. The essence of the method is of course to put together two identical triangles to make a rectangle. This can be done either physically, by using a second triangle; or by drawing.

4. If the triangles are put with a pair of the smaller sides together, the result will be a parallelogram or an isosceles triangle, depending on the relative orientation of the right triangles. This is a little harder, and anticipates Stage (b).

420

Stage (b) Scalene triangles This repeats steps 1 to 4 of Stage (a), this time with scalene triangles. In

this case the result will always be a parallelogram, and by now they are well equipped to deal with these.

5. The area of the original triangle is half that of the parallelogram. In the diagram above, this is half one of the length of one of the parallel sides multiplied by the distance between them, shown by the dotted line.

Steps6to9shouldnotbetakenuntilthefinalpartofthemethodiswellestablished.

6. In the original triangle, this dotted line is called an altitude of this triangle. 7. Two other altitudes can be drawn, each perpendicular to one of the sides. These

are perpendicular to opposite sides of the two other parallelograms which can be made in the way described in above. The children should verify this.

8. So in terms of the original triangle, the area can be calculated by multiplying the length of any side by the altitude perpendicular to this side, and halving the result.

9. I suggest that the use of an obtuse-angled triangle is best dealt with as a teacher-led discussion. Here, two of the altitudes require dotted extensions of their opposite sides.

Meas 2.5 Other shapes convertible to rectangles (cont.)

421

Activity 3 Area of a circle [Meas 2.5/3]

An activity for children to work on initially by themselves, followed by a teacher-led discussion. Its purpose is to introduce them to the method for calculating the area of a circle.

Materials For each child: • A circle on paper, for cutting out.* • A sheet of coloured paper. For the group: • Coloured felt-tips. • Several glue sticks or equivalent. * A suitable one is provided in the photomasters.

What they do 1. They cut out their circles, fold them in half, open out again and colour one half (that is, one semicircle).

2. They now fold again into halves, then quarters, then eighth-parts. 3. They open out, and cut along the creases to get eight pie-shaped pieces. These

are called sectors of the circle.

4. These sectors are then pasted on the sheet of coloured paper with the white and coloured sectors pointing in alternate directions, as shown.

1. What shape does this look like? (A parallelogram, with one pair of sides straight and the other pair wavy.)

2. How long is each of the wavy sides? (Following the curve, it is half the circum-ference of the original circle.)

Meas 2.5 Other shapes convertible to rectangles (cont.)

Suggested outline for the discussion

half the circumference

radi

us

422

3. What is the distance between them? (It depends where we measure. If we measure from the vertex to the arc of a sector, it is a radius of the original circle (and of each individual sector).

4. So the area of this ‘nearly-a-parallelogram’ is approximately? (Half the circum-ference multiplied by the radius.)

5. And the area of the original circle? (The same.) 6. How could we make it closer to a parallelogram? (By folding into sixteenth-

parts and using these. Some children may like to try this.) 7. Would this change its area? (No.) 8. Could we make it closer still? (Yes, by using smaller and smaller parts. In this

case we could not do it by folding, but to begin with we could draw them.) 9. Could we go on making it closer and closer to a parallelogram, so close that the

difference was too small to matter? (Yes; even when it became too hard to draw, we could continue in our imagination.)

10. And the area of this ‘can’t-tell-it-from-a-parallelogram’ is? (No change; it is still what we said before.)

11. So the area of the original circle is? (Half the circumference multiplied by the radius.)

Note Later, when they have learned about π, they will be able to use this to calculate the length of the circumference from the radius, and thereby to arrive at another form of the result given above.

By the end of this topic, we have come a long way from counting unit squares with the help of a centimetre grid. Using our knowledge of how to calculate the area of a rectangle, we develop in succession methods for calculating areas of parallelo-grams, triangles, and finally circles, the last of which doesn’t look anything like a rectangle. These are activities in pure mathematics, and another nice example of mathematical creativity — using our existing knowledge to create new knowledge, while testing always for consistency with our existing knowledge.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 2.5 Other shapes convertible to rectangles (cont.)

Discussion of activities

423

Meas 2.6 LARGER UNITS FOR LARGER AREAS (square metre, hectare)

Concepts (i) That larger units are need for measurement of larger areas. (ii) Square metres, hectares.

Ability To calculate areas in these units.

These are straightforward expansions of the original concept of a square centimetre as a measure of area.

Activity 1 Calculating in square metres [Meas 2.6/1]

A teacher-led discussion, after which the children work by themselves, and compare results.

Materials For each child: • Example sheets 1 & 2, with dimensioned drawings of areas to be calculated in

square metres. These should be similar to those which they have already en-countered in square centimetres. Sheet 1 has only whole number dimensions, but sheet 2 has some of the dimensions in metres and tenth-parts.*

• Pencil and paper. • For sheet 2, a calculator. * Suitable example sheets are provided in the photomasters. The second of these

includes one new shape, a trapezium.

1. Suppose that we want to measure larger areas, such as that of a classroom, or a garden, or a swimming pool. Would square centimetres by a good unit for doing this? (No. The numbers would be too big, and we do not want to measure to an accuracy of 1 cm. In the case of a garden, we couldn’t even if we wanted to.)

2. So what could we use instead? (The answer we want to end up with is a square metre. If anyone suggests square decimetres, this too is a good answer. For measuring the dimensions of (say) a room, though, centimetres would usually be too accurate a measure; metres (in whole numbers) would not be accurate enough. However, while a square decimetre seems to me quite a sensible unit, it is not in general use. In the case of a room, the usual practice would be to measure in metres, or metres and tenth-parts (e.g., 7.3 metres by 5.7 metres), and in the second case to use a calculator for multiplying.)

3. In square metres, what would be the area of a rectangle 3 metres by 5 metres? 6 metres by 4 metres? 30 metres by 90 metres?

What they do 1. Each child has a copy of example sheet 1. They all calculate the areas of each of these examples (on separate sheets of paper, not the example sheet).

2. They compare results, and discuss any differences until they agree on the result. 3. Steps 1 and 2 are repeated with example sheet 2.

Suggested outline for the discussion

Discussion of concepts

424

Activity 2 Renting exhibition floor space [Meas 2.6/2]

An activity for children to work on individually or in pairs. Its purpose is to give practice in calculating area in square metres.

Materials • Dimensioned plans of two or more exhibition halls, showing floor spaces for individual stands, and the daily rental charge for these per square metre. Enough copies are needed for each child to have one or the other of these plans.*

• Pencil and paper for each child. * Suitable dimensioned plans are provided in the photomasters.

What they do 1. Each child has one of these plans, and calculates (a) the rental charge for each space at $ ___ per square metre, and (b) the total daily income for the whole hall.

2. When they have finished, children with copies of the same plan may compare results and discuss any disagreements. If these cannot be resolved, they seek help from their teacher.

Activity 3 Buying grass seed for the children’s garden [Meas 2.6/3]

This is similar to Activity 2, with a different embodiment. It is harder, and may be better taken as a cooperative activity for two or three children.

Materials For each child: • Children’s garden plan, see Figure 48.* • Children’s garden checklist.* • Pencil and paper. * Provided in the photomasters

What they do 1. Each child has a copy of the plan, which represents a garden of an English stately home. Originally this was owned by a rich family, but it is now open to the public, so children use the garden for playing in. The ornamental pool is now used for paddling, the yew trees for hide-and-seek, the summer house for quiet reading. Older children with their teachers sometimes measure the fountain.

2. As a result it needs to be re-sown with extra hard wearing grass. 3. Each child is asked to calculate the cost of doing this, given that the cost of grass

seed is $11.00 per 1 kilogram box, and each kilogram is enough for 20 square metres.

4. When all have finished, they compare results and discuss any disagreements. 5. (Optional) They are then told that the seed can also be bought in boxes of

various sizes, costing $6.00 for a 0.5 kg box, $19.00 for a 2 kg box, and $40.00 for a 5 kg bag. What would the cheapest way to buy it?

Note These prices were correct when going to press, but may need updating.

Meas 2.6 Larger units for larger areas (square metre, hectare) (cont.)

425

Figure 48 The children’s garden

Meas 2.6 Larger units for larger areas (square metre, hectare) (cont.)

30 m

42 m

Fountain

Hedge

Yew trees

Grass

Grass Grass

Seats for parents

Summerhouse

Flowerbed

Flowerbed

Steps

Pool

6 m

4 m 3 m

14 m 7 m

2m

16 m

6 m

8 m

6 m

6 m

Fountain circumference: 22 m

2m

2m

2m

4 mStatue

Fountain

10 m2m

6 m

426

Activity 4 Calculating in hectares [Meas 2.6/4] The purpose of this activity is to introduce children to the hectare as a unit for

measuring large areas. It is introduced by a teacher-led discussion, after which the children work individually or in pairs. Finally they compare results and discuss any disagreements.

Materials • For each child or pair, a copy of the example sheet provided in the photomasters. • Pencil and paper for each child.

1. Explain that for measuring land areas a unit often used is the hectare. This is the area of a 100 metre square.

2. So has a 200 metre square an area of 2 hectares? (No. They should make a rough drawing, from which they can see that the area is 4 hectares.) And the area of a 300 metre square? Of a 400 metre square?

3. Ask for some oblongs which would have an area of 1 hectare. (Examples: 200 m by 50 m, 20 m by 500 m, 40 m by 250 m, 80 m by 125 m, . .

. .) 4. It may help to imagine the size of a hectare if we know that a lawn tennis court

is 0.03 hectares, a soccer field can be as large as 1.08 hectares (max.) or as small as 0.41 hectares (min.), and a regulation Canadian football field, including the end zones, is about 0.87 hectares.

5. How many square metres are there in a hectare? (10,000) 6. What is the area of a rectangular plot of land 160 metres by 250 metres? (40,000

square metres, which is equal to 4 hectares.) And of one 200 metres by 120 me-tres? (24,000 square metres, which is equal to 2.4 hectares.)

What they do They should now be ready to calculate the areas of the plots of land shown in the example sheet. In shape these are rectangles, parallelograms, triangles, and circles. As in previous activities they work individually, compare results, and discuss any disagreements.

Activity 5 Buying smallholdings [Meas 2.6/5]

This activity is for further practice at calculating in hectares. Some of the shapes are less straightforward than those in Activity 4.

Materials For each child • Example sheets 1 (easier) & 2 (harder).* • Pencil and paper • For sheet 2, a calculator. * Provided in the photomasters

What they do 1. Each sheet represents a number of adjacent plots of land which are to be sold as smallholdings. Prices are based on their areas, which therefore need to be calculated.

2. As before they work individually, compare results, and discuss any disagreements.

Meas 2.6 Larger units for larger areas (square metre, hectare) (cont.)

Suggested outline for the discussion

427

In this topic, they encounter requirements for which their original unit of area, the square centimetre, is too small. The concept of measuring area should however have been well formed and consolidated in the five topics which come before, and thus be ready for expansion. This is Mode 3 concept building — expansion of existing concepts. Mode 3 testing (consistency with existing knowledge) is built in, since the new units are closely related to the old. Mode 2 building takes the form of communication from their teacher, and Mode 2 testing is provided by plenty of discussion, both teacher-led and within groups.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activities

Meas 2.6 Larger units for larger areas (square metre, hectare) (cont.)

Buying smallholdings [Meas 2.6/5]

428

Meas 2.7 RELATIONS BETWEEN UNITS

Concept That of numerical relations between different units of area.

Abilities To convert in both directions between square centimetres, square metres, and hectares.

One example of this concept has already been encountered in the previous topic, in the relationship between square metres and hectares. In this topic we make the concept explicit, expand it, and organize some of the particular cases in the form of a table.

Activity 1 “What could stand inside this?” [Meas 2.7/1]

A teacher-led discussion for a group of children, possibly the whole class. Its pur-pose is to consolidate their notions of the relative sizes of the units listed above and relate these to their everyday experience. In this activity we also include the ‘are’ (pronounced as in area), since it is from this that the hectare is derived. However, I do not think that the ‘are’ is widely used, so it is not included in any of the other activities.

1. Draw on the blackboard a metre square, and inside it stick a little piece of paper 1 cm square. Have ready a piece of cord 10 metres long.

2. Start with the centimetre square. What could stand inside this? (A fly, a flea, a wood louse, . . . )

3. Next, the metre square. What could stand inside this? (One of us, a bird, a squirrel, . . . )

4. Stretch out the 10 metre cord, and ask them to imagine a square with this as side. The area of this is called an ‘are.’ What could stand inside this? (Several horses, a medium sized van, a small house, . . . )

5. Ask them to imagine a square with side 100 metres. (A Canadian football field is 100 metres long.) What could stand inside this? ( A large herd of cattle, a department store, a fleet of trucks, . . . )

Activity 2 Completing the table [Meas 2.7/2]

An activity for children to do individually. Its purpose is to summarize what they have learned about the relations between units of area.

Materials • Pencil, paper, and ruler for each child.

What they do 1. Each child copies and completes the table illustrated below. 2. When completed, they compare results and discuss any disagreements. If these

cannot be resolved, they consult their teacher.

Discussion of concept

Suggested outline for the discussion

429

TABLE

square square metres hectares centimetres

= 1 =

--- = = 1

1 = ---

(The dashes indicate spaces which it would not make much sense to fill.)

This is what they should get.

TABLE

square square metres hectares centimetres

10 000 = 1 = .0001

--- = 10 000 = 1

1 = .0001 ---

Activity 3 “It has to be this one.” [Meas 2.7/3]

An activity for children to work at individually. Its purpose is to give practice in estimating what would be a sensible answer, so that they can check that the result of using a calculator makes sense. Here it also gives practice in using what they have already learned about calculating area.

Materials • One or more example sheets.* • Pencil and paper for each child. * Two suitable ones are provided in the photomasters.

What they do 1. The example sheet represents a page of diagrams as they come from the printer. The dimensions form part of the drawings, but the areas have been typeset sepa-rately and appear at the bottom.

2. These areas were worked out by calculator, but for this exercise the children do not have calculators.

3. But if the drawing is of a rectangle, (say) 71.4 metres by 185 metres, then out of the following, they should be able to choose the right one.

12 500 sq cm 1 260 sq m 1.3 hectares 4. A good way to do this would be to round the 71.4 down to 70 metres, and the

185 up to 200 metres. These can be multiplied without a calculator: result 14 000 square metres, which is equal to 1.4 hectares. 5. So for this rectangle it has to be the last figure, 1.3 hectares. 6. When they have finished, they may compare results and discuss any disagree-

ments. If these cannot be resolved, they seek help from their teacher.

Meas 2.7 Relations between units (cont.)

430

The first activity is for relating their newly-acquired mathematical concepts to their everyday knowledge, that is to say, relating their school learning to the world outside school. By assimilating their mathematical knowledge to a wider schema, they are broadening their understanding. The second activity is for organizing and consolidating their knowledge of the numerical relationships between units, so that they have these available in the form of ready-to-hand methods. In the last activity they are looking at a variety of possible answers made avail-able to them and deciding which one makes sense in relation to what they already know. This is Mode 3 testing, and is followed by Mode 2 testing with others at their table.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 2.7 Relations between units (cont.)

Discussion of activities

“It has to be this one.” [Meas 2.7/3]

431

Meas 2.8 MIXED UNITS

Concepts (i) That different units may sometimes be used for different dimensions of the same object.

(ii) That there is no recognized meaning for multiplying these as they stand.

Ability To calculate areas correctly in these conditions.

This is a straightforward development from their existing knowledge that different units are used for different jobs. Here, they are used for different parts of the same job, as described in Activity 1.

Activity 1 Mixed units [Meas 2.8/1]

An activity for children to children to do individually, introduced by a teacher-led discussion. Its purpose is to enable children to avoid being confused by mixed meas-ures, and to calculate correctly under these conditions.

Materials For each child: • Example sheets 1 & 2.* • Pencil and paper. • Calculator for the harder examples. For yourself: • An unopened package of paper towels. * Provided in the photomasters.

Note: Readers who have available a copy of Mathematics in the Primary School mightfinditinterestingheretoreadthethirdparagraphonpage5,whichbe-gins “If the teacher asks . . .”

1. Show them the roll of paper towels, and tell them the measurements given by the manufacturer. A typical example is 23 centimetres wide, 20 metres long.

2. Ask how they would calculate its area. 3. If someone wants to start by multiplying 23 by 20, the next question is: 4. What would the unit of measurement for the result be? The result can’t be 460

square metres, which is much too big; for this, the roll would have to be 23 metres wide. It can’t be 460 square centimetres, because we would get this if the roll were only 20 cm long. And an oblong 1 cm by 1 metre is not an accepted unit. (It could be used for a unit of area, but it never is because it would be a very inconvenient unit.)

5. We want our answer in either square centimetres or square metres; obviously, not in hectares.

6. For this, length and breadth would have to be either both in centimetres or both in metres, so one would have to be converted. Let’s try both ways.

7. 23 cm by 2 000 cm. Result, 46 000 sq cm. 8. .23 m by 20 m. Result, 4.6 sq m. 9. Which gives a better idea of the total area? There is no right or wrong answer to

this, though I, myself, prefer the second.

Discussion of concepts

Suggested outline for the discussion

432

What they do 1. They are now ready to work through the easier examples. When they have done these individually, then as usual they compare results and discuss any disagree-ments. If these cannot be resolved, they consult their teacher.

2. Likewise with the harder examples.

The teacher-led discussion is a good example of Mode 3 building — extrapolation of existing knowledge and methods to meet the requirements of a new situation. The teacher-led discussion leads the children in the right direction for doing this. Mode 3 testing, whether the incorrect results which children have been known to give make sense in terms of what we know already, and Mode 2 testing, whether an oblong 1 centimetre by 1 metre would ever be agreed on as a sensible unit of area, complete the process of concept formation. It is then consolidated by suitable examples.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of activity

Meas 2.8 Mixed units (cont.)

Which of these can hold more? [Meas 3.4/1]

433

[Meas 3] Volume and capacity

Meas 3.5 MEASURING VOLUME AND CAPACITY USING NON-STANDARD UNITS

Concept The capacity of a container as distinct from the volume held at a particular time.

Ability To order containers according to capacity or volume held, using non-standard units.

activity 1 putting containers in order of capacity [Meas 3.5/1]

An activity for a small group of children working together. Its purpose is to extend their abilities to measure, using a non-standard unit of a different kind, and to intro-duce the term ‘capacity.’

Materials • An assortment of glasses, jugs, jars, etc., no two alike and varying as widely as possible in height and width.

• For each child, an identical container smaller than any of these, such as an egg cup.

• Water.

What they do 1. Explain that their task is to arrange the containers in order of how much they can hold when full. This is called their capacity.

2. To begin, each child takes one of the containers. 3. After that, it is for them to work out the best method. That each has an egg cup

gives a clue. 4. One good method is, of course, to use the egg cup as a non-standard unit, record-

ing the number of these that it takes to fill one’s own container. 5. If there are more containers, steps 2 to 4 are repeated. Those not thus engaged

can use the results already obtained to begin putting in order those already measured.

activity 2 “Hard to tell without measuring” [Meas 3.5/2]

A continuation of Activity 1 for two teams that underlines the need for measurement, and that gives children experience applying their concept of volume more generally.

Materials • As for Activity 1, except that the containers are now lettered.

What they do 1. Using their egg cups, team A pours a different volume of water into each of the containers. The amount in each container is recorded. Team B does not watch while this is done.

2. Team B now tries to arrange the containers in order of how much water is in them. 3. While team B is doing this, team A writes the letters in order of how many egg

cups full were poured into the containers. 4. These two orders are now compared. 5. Steps 1 to 4 may now be repeated with the teams changing roles.

These two activities, like the earlier ones in Volume 1, extend the children’s con-cepts and abilities by giving them problems to solve with physical materials. The first one concentrates on capacity, and the second on volume in general.

discussion of concept

discussion of concepts

434

Meas 3.6 STANDARD UNITS (LITRES, MILLILITRES, and kILOLITRES)

Concepts (i) The litre as a standard unit for measuring capacity. (ii) The millilitre as a standard unit for relatively precise capacity measurement. (iii) The kilolitre as a standard unit for measuring relatively large capacities.

Abilities To estimate, measure, and compare the capacities of containers using litres, millilitres, and kilolitres.

While non-standard units can be useful for comparing the capacities of two or more containers directly, standard units are essential when it comes to making purchases. For every day buying or selling of liquids, the litre (L) and the millilitre (mL) are usually the units of choice. The following activities focus on the sizes and usefulness of these units of measure.

activity 1 can you spot a litre? [Meas 3.6/1]

An activity for the whole class to draw attention to the relative size of a litre, to the many ways in which this unit is used, and to the variety of shapes that can be found that will hold a litre.

Materials For each group: • A collection of containers brought in beforehand by the children. • A variety of other containers, including some that hold exactly one litre, and, if

available, a clear plastic cube open on one side and with internal measurements 10 cm by 10 cm by 10 cm (i.e., with a capacity of 1000 cubic centimetres or 1000 millilitres or 1 cubic decimetre or 1 litre . . . and occupying a volume of little more than 1000 cubic centimetres).

What they do 1. Well before the class activity, ask the children to bring empty containers from home which they think would hold about the same amount of liquid as, for example, a 1 L milk carton (available for the children to view). [They will likely bring in milk containers, apple juice boxes, ice cream containers, etc.]

2. To begin with, each child takes one of the containers and determines whether it has the capacity of one litre, or more, or less.

3. Then they demonstrate to the others in the group which of the containers have the same capacity (even though they are perhaps very different in appearance)

. . . and which, in particular, can hold exactly one litre. 4. When they are satisfied which containers have a one-litre capacity, the children make

signs like the following: ‘less than a litre (L),’ ‘1 litre (L),’ ‘more than a litre (L).’ 5. They predict the capacities of each of the whole collection of containers, placing

them with the appropriate sign. 6. They then check their predictions by using the one-litre containers. Finally,

they describe what they have done. Some may like to point out what they have noticed about the 10 cm by 10 cm by 10 cm container (if available).

discussion of concepts

435

activity 2 “Special: Small lemonade, 10¢” [Meas 3.6/2]

An activity for a group of six children. Its purpose is to call attention to the need for a smaller standard measure for volume or capacity than the litre.

Materials For each group: • Paper cups of two sizes with shapes that make it difficult to judge by eye which

has the greater capacity (e.g., cone-shaped versus nearly cylindrical). • A jug of water. • A plastic beaker graduated in mL. • The litre containers from Activity 1. • Clear containers on which the height of liquid within can be marked easily.

What they do 1. Two of the children take the part of lemonade stand operators. One runs the “Good-for-you” stand and the other the “Tasty Aide.” They each prepare a sign: “Special: Small Lemonade, 10¢.” Unknown to the operators (at least initially), the ‘small’ cups supplied to one stand have a larger capacity than those supplied to the other.

2. The rest of the children act as customers. While the owners are preparing their signs, the customers (who have heard of the ‘specials’) write on a sheet of paper something like: “Good-for-you or Tasty Aide . . . where shall I buy some lemonade?”

3. The customers will, of course, want to find out which stand offers the best buy. They can do this by pouring small cups of lemonade into a litre container but this would require buying a lot of lemonade and would likely take a lot of time. Can they think of another way to find out which is the better buy?

4. When they think they have discovered an answer, they write a description of what they have found and compare with the other customers and the lemonade sellers.

Note Some may pour the cup of lemonade into a container, mark the height, pour the liq-uid out, and then pour the contents of the second cup into the same container, com-paring the heights. Others may decide to use the graduated mL beaker. Advantages and characteristics of the beaker are discussed. Reading the scale is similar to using a ruler to measure length, but the children are likely to want to experiment to confirm that the markings on the beaker scale correspond to liquid volumes in mL and L.

activity 3 making and tasting (accuracy in the kitchen) [Meas 3.6/3]

A small group activity in which the result of any inaccuracies is likely to be very obvious! Some children may have had experience baking with grandparents or par-ents who successfully use a “pinch of this” and “about that much” of something else. Explain that when we are learning to bake, it is best to use exact amounts.

Materials • Recipes with measurements in L and mL. There are many ‘no-bake’ children’s

recipes available that could suit the purpose admirably. • The necessary ingredients for such treats as ‘Crunchy peanut butterballs,’ ‘cocoa

rolled-oat cookies,’ ‘no-bake chocolate cookies,’ ‘banana bread,’ ‘yogurt dip,’ . . .

Suggestions The purpose of the activity is to provide experience with simple recipes while be-coming more familiar with litres and millilitres. For younger children, one could use recipes which do not require very precise measurements, whereas older children could work with measurements such as 5 mL vanilla, 60 mL vegetable oil, etc.

Meas 3.6 Standard units (litres, millilitres, and kilolitres) (cont.)

436

activity 4 Race to a litre [Meas 3.6/4]

An activity for 2 to 4 players to practise measuring mL in a game situation.

Materials • A 1-L container for each player. • A plastic beaker graduated in mL. • A large container of water. • A die marked: 25 mL, 50 mL, 100 mL, 150 mL, 175 mL, and 250 mL.

What they do 1. Players roll the die in turn and measure the amount of water indicated by the die. 2. Each observes carefully as the others measure the water into their own containers. 3. The first player to reach 1 litre or more is the winner.

activity 5 completing the table of units of capacity or volume [Meas 3.6/5]

An activity for children to do individually. Its purpose is to summarize what they have learned about the relations between litres and millilitres and to introduce a third standard unit for measuring capacity or volume: the kilolitre.

Materials • Pencil, paper, and ruler for each child.

What they do 1. Each child copies and completes the table illustrated below. 2. When completed, they compare results and discuss any disagreements. If these

cannot be resolved, they consult their teacher. It might be interesting to let them try to name the mystery unit, the ‘?,’ and to suggest where it might be used in-stead of litres (e.g., amount of water in a swimming pool or hot tub).

TABLE

millilitres litres ? mL L ?? = 1 = .001 = = 1 1 = =

TABLE

millilitres litres kilolitres mL L kL 1 000 = 1 = .001 1 000 000 = 1 000 = 1 1 = .001 = .000 001

These activities bring out the everyday, practical usefulness of measuring volume and capacity in standard units based on the litre. They provide opportunities for connecting home experiences with classroom activities.

This is what they should get.

discussion of activities

Meas 3.6 Standard units (litres, millilitres, and kilolitres) (cont.)

437

[Meas 4] Mass and weight

INTRODUCTORY DISCUSSION

Mass and weight are closely related, and often confused for reasons which will emerge in this discussion. Though we do not want to cause difficulties for children in the early stages by making too much of this distinction, it is easier to understand something which is accurate than something which is confused. So the object of the present discussion is to get the relations between mass, weight, and inertia as clear as possible in our own minds, and to outline the ideas which underlie the ap-proach I have taken in this network. At an everyday level, we can say that a mass is anything which has weight. So a stone, a book, a house, a person are masses, whereas a day of the week, a poem, a joke, are not. This is much the same at a scientific level, since weight is the gravi-tational attraction of the earth on all other masses. Every mass attracts every other mass with a gravitational force; but this is very small, unless at least one of the masses is very large, which is the case with the earth. Weight is one of the characteristics of mass, the other being inertia. On the moon, the same mass would weigh less, and at some location in between, where the gravitational pull of the moon was equal and opposite to that of the earth, it would be weightless. But in all these places its inertia would be the same. When we push a car in neutral on smooth level ground, it is its inertia which requires a hard push to get it started, and an equally hard push backward to stop it. The forces required would be much less for a bicycle, and much more for a railway car, though the frictional resistance is small, even for the last of these. This would still be the case at each of the three locations described, though the weights would now be different. Both weight and inertia are common everyday experiences. It is the weight which makes our arms tired when we carry a child; it is inertia which causes hurt when a child falls over. However, weight is much easier to measure (e.g., by bath-room scales). Given that the weights of two bodies are proportional to their mass-es, if both are the same distance from the centre of the earth, which for everyday situations they are, then the easiest way to measure a body’s mass is to compare its weight with that of a standard mass. So for both these reasons, we shall approach the concept of mass through the experience of weight, and the measurement of mass by the measurement of its weight. In this network, the hardest concepts come at the beginning and I would not expect children to grasp them in the form outlined above. My aim in the activities which follow is not to sweep anything important under the carpet, but to keep it visible though not necessarily fully discussed. The children will then, I hope, not have anything to unlearn later.

438

discussion of concepts

Meas 4.4 MEASURING MASS BY WEIGHING, NON-STANDARD UNITS

Concept Weighing as a way of measuring mass.

Abilities (i) To measure the mass of a given object in terms of non-standard units. (ii) To compare the masses of two or more objects, using non-standard units.

Mass and weight are not the same, but, because they are very closely connected, when we compare weights of objects we are also comparing their masses. The need for standard units has already been shown in the LENGTH network, Meas 1. In view of its importance, a similar sequence is followed here, in a more condensed form. The present topic is used to underline the utility of weighing, by introducing it in a problem-solving situation (Activity 1). Activity 2 again calls attention to the importance of standard units. When measuring length, we combine units by putting them end to end. For area, we need units which tessellate, to cover a surface with no gaps and no over-lap. Combining unit weights turns out to be straightforward, since if we put them in the same pan of a balance the gravitational forces on all of them are combined into a single downward force.

activity 1 Problem: to put these objects in order of mass [Meas 4.4/1]

A group problem-solving activity. Its purpose is to lead children to the concept of weighing as a means of solving a problem which has been presented to them.

Materials • An assortment of eight or more objects, most of them clearly distinguishable in weight, but with (say) three of the same weight. All should look different but with as little connection as possible between appearance and weight. So there should be small heavy objects and large heavy objects, small light objects and large light objects (i.e., ‘light’ and ‘heavy’ . . . relative to the other objects). All should be light enough for children to lift easily.

• A suitable set of unit objects in a transparent bag (see Notes below). Collectively, these must be heavier than the heaviest objects in the assortment just described.

• A balance. • Pencil and paper • Two envelopes containing hints (see steps 4 and 6 below and the photomasters). Notes (i) It is not easy to think of unit objects for mass which can readily be found in

schools and which fulfil the requirements of being all of the same mass, fairly heavy for their size (the plastic cubes so readily available are much too light) and about 50 to 100 grams in mass so that we can weigh up to 1 kilogram with up to 20 of them. After much thought, my suggestion is that you buy a bag of bolts (the metal kind which screw on to nuts) from a hardware store. It should be possible to find some of suitable size for our present purpose. If you think of something better, I hope that you will share it with me and other teachers.

(ii) You will need another bag of bolts (or whatever you choose) of slightly smaller mass for Activity 2, so I suggest that you read this activity also before shopping.

439

What they do 1. Give them all the materials listed, with the bag of bolts included among the others.

2. The problem is to put all of these objects in order of mass. 3. To do this by comparing them in pairs would be possible, but laborious. If they

start out this way, it might be worth letting them find out how laborious this is, before asking if they can think of a better way.

4. If they find themselves stuck, they may ask you for an envelope containing a hint.

5. The first hint reads: “You may open the bag of bolts.” This suggests that they may have some special significance, and implies the possibility of using one or more of these separately.

6. If after a time they are still stuck, they may ask for another hint. This reads: “Choose one of the objects, and see how many bolts it takes to balance this.”

7. This, of course, as good as tells them the solution. After this, it is straightfor-ward to order the numbers of bolts equal (or approximately equal) in mass to each object, and put these numbers in order.

8. (Optional) Ask them to write a short account of the ways they tried, and how measurement was found to be the key to success.

activity 2 honest hetty and Friendly Fred [Meas 4.4/2]

An activity for a small group of children. Its purpose is to call attention to the need for standard units, and thus prepare for the next topic.

Materials • Twelve or more large potatoes (six or more for each stall). • Two balances. • The bag of bolts used in Activity 1, for Honest Hetty. • A similar bag, in which the bolts are like the others but lighter, for Friendly Fred. • Notices as illustrated below.* * Provided in the photomasters.

Meas 4.4 Measuring mass by weighing, non-standard units (cont.)

COMe tO hOnest hettYFOR YOUR

deLiCiOUs BaKed POtatOesChOOse YOUR Own

and PaYOnLY One Cent

FOR eaCh BOLt Mass

COMe tO FRiendLY FRedFOR YOUR

FaVOURite BaKed POtatOesChOOse YOUR Own

and PaYOnLY One Cent

FOR eaCh BOLt Mass

440

What they do 1. Honest Hetty and Friendly Fred are baked-potato sellers in the same mall. They lay out their stalls, with potatoes, balance, and the bolts which will be used for weighing, with their notices prominently displayed. The other children act as customers.

2. Some customers are served by Honest Hetty and some by Friendly Fred. 3. After several purchases from the two stalls, several of the more enterprising

customers get together to find out which of the two stalls gives the most for the money, or whether they are both the same. They need to work out a method for doing this.

4. Finally a report of the findings is given.

Both of these activities involve plenty of social interaction and discussion. The practical usefulness of weighing is brought out in Activity 1, and the social im-portance of reliable measures is shown in Activity 2. Both of these, especially the latter, are small scale counterparts of applications of weighing which are important in everyday life.

discussion of activities

Meas 4.4 Measuring mass by weighing, non-standard units (cont.)

OBSERVE AND LISTEN REFLECT DISCUSS

Honest Hetty and Friendly Fred [Meas 4.4/2]

441

Meas 4.5 STANDARD UNITS (kILOGRAMS)

Concept The kilogram as a standard unit for measuring mass.

Abilities To estimate, measure, and compare the masses (weights) of objects using kilograms.

As seen in the last topic, non-standard units may be useful for comparing the masses of two or more objects directly, but when it comes to communicating in the world of commerce, standard units are essential. For every day buying or sell-ing by weight (mass), the kilogram (kg) is often the unit of choice. The following activities focus on what a kilogram is and its usefulness.

activity 1 Making a set of kilogram masses [Meas 4.5/1]

A small group activity to familiarize children with the weight of a kilogram and to produce an economical set of kilogram masses for the following activity.

Materials For each group: • A sturdy equal-arm, two-pan balance. • A standard one kilogram mass (or 1 kg substitute). • Seven empty, resealable containers which can hold up to 1 kg of sand (e.g., 500

mL milk cartons or small self-closing plastic freezer bags). [An alternative to the seven 500 mL milk cartons: one 500 mL, one 1 L and one 2 L carton]

• At least seven kilograms of sand, Plasticine, or other suitable material. • A tablespoon or equivalent. • Masking tape. • Marking pen.

What they do 1. A single standard one-kilogram mass can be used to make a set of objects for weighing up to 7 kilograms, as follows.

2. Each child carefully spoons sand into a resealable container on one pan of the balance until the standard kilogram mass on the other pan is balanced exactly. This is repeated until the group has seven newly-made ‘kilograms.’

3. Six of the newly-made kilograms are then taped together in pairs. (Alternative-ly, the newly-made 1-kg mass can be placed on the pan with the standard 1-kg mass so that a 2-kg mass can be made by placing sand in a suitable container.)

4. Finally, two of the pairs of the ‘new’ 1-kg masses are taped together, resulting in a home-made set of 1, 2 and 4-kilogram masses. (Or the alternative described in step 3 can be taken a step further to make a 4-kg mass.)

5. Question: What masses can be balanced using these 1, 2 and 4-kilogram masses? 6. The children are then asked to find objects that have masses that they think can

be weighed using their home-made weights and the balance. For each, they are asked to record an estimated weight (e.g., “more than 2 kg”), and then to use the balance and the 1, 2, and 4 kilogram masses to check their estimates.

discussion of concept

442

activity 2 Mailing parcels [Meas 4.5/2]

An activity for about four children to find out the cost to mail a parcel when its mass, rounded up to the nearest kilogram, is known.

Materials • A sturdy equal-arm, two-pan balance. • At least one set of 1, 2, 4-kilogram masses (as made in Activity 1). • Suitable objects to wrap for ‘mailing’ (e.g., textbooks) to produce packages with

masses in the range 1 to 7 (or more) kilograms. • Plain wrapping paper. • A ‘small packages’ postal rates chart.* • Marking pen. * A sample is provided in the photomasters.

What they do 1. Ask each of the children to wrap a parcel (or parcels) to send to someone they know (arranging for a suitable range of masses, if possible).

2. One child acts as postal clerk and each of the others presents a parcel for mailing. The clerk ‘weighs’ the parcel, using a marking pen to record its weight, to the next higher whole kilogram, on the parcel. Then, a chart like the following is used to determine the postage needed (typical rates within the same postal code area).

Mass Postage (up to and including) 1 kg $2.80 2 kg $2.80 3 kg $3.16 4 kg $3.52 5 kg $3.88 6 kg $4.24 7 kg $4.60 8 kg $4.96 9 kg $5.32

3. Each child takes a turn as the postal clerk. Any differences in mass or postal rate for a given package should be discussed.

Both of these activities provide Mode 1 experience with the kilogram as a unit of mass as well as Mode 2 discussions regarding accuracy and applications. The 1, 2, 4, . . . pattern and ideas of using materials of different density for ‘filler’ for the homemade kilogram objects may lead to some interesting Mode 3 extrapolations.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 4.5 Standard units (kilograms) (cont.)

discussion of activities

443

Meas 4.6 THE SpRING BALANCE

Concept The spring balance as a convenient device for measuring mass.

Abilities To measure and compare the masses of objects in kilograms using a spring balance.

The “spring balance” is not a balance at all and is often referred to, more appro-priately, as a “spring scale.” It consists of a pan or hook that hangs from a spring or a pan or platform that rests on a spring. The object being weighed stretches or compresses the spring, moving a pointer relative to a scale to indicate the mass of the load. To begin with, the children should have the opportunity to verify that the scale readings agree with the values that they have given their homemade masses.

activity 1 Checking a spring balance [Meas 4.6/1]

An activity for two children to check the reliability of scale balance readings. Materials • A spring balance suitable for measuring mass in the 1 to 10 kg range. • At least one set of 1, 2, 4-kilogram masses (as made in Meas 4.5/1).

What they do 1. The children cooperatively verify whether or not the spring scale readings agree with the masses of the homemade kilogram masses.

2. If there are discrepancies, appropriate adjustments should be made before pro-ceeding to the next activity (including ‘topping up’ any sand spilled).

activity 2 Mailing parcels (spring balance) [Meas 4.6/2]

A repeat of Meas 4.5/2, finding the cost to mail a small parcel when its kilogram mass is found using a spring balance.

Materials • A spring balance suitable for the 1 to 10 kilogram range. • The wrapped objects used in Meas 4.5/2. • The ‘small packages’ postal rates chart used in Meas 4.5/2.* • Marking pen. * A sample is provided in the photomasters.

What they do 1. As in Meas 4.5/2, one child acts as postal clerk and each of the others presents a parcel for mailing. The clerk weighs the parcel using the spring scale, noting the whole kilogram reading next above the position of the pointer. This is compared with the mass previously marked on the parcel. A chart like the one used in Meas 4.5/2 is used to confirm the postage needed.

2. Each child takes a turn as the postal clerk. Any differences in mass or postal rate for a given package should be discussed.

Though Activity 2 only requires whole-kg measures, the spring scale very likely provides for measurement of masses between whole kilograms, which can be used to set the stage for the introduction of ‘grams’ in the next topic.

OBSERVE AND LISTEN REFLECT DISCUSS

discussion of activities

discussion of concept

444

Meas 4.7 GRAMS, TONNES

Concepts (i) The gram as a standard unit for relatively precise mass measurement. (ii) The tonne as a standard unit for measuring relatively large masses.

Abilities (i) To estimate, compare, measure, and order the masses of objects, to the nearest 10 grams.

(ii) To appreciate the size of a tonne mass and applications in which it is used.

Scale balances suitable for use by elementary school children are not likely to be sensitive enough for reliable comparison of masses that differ by less than 10 grams. Some appreciation can be gained for the small size of a single gram (4 pennies have a mass of about 10 g) and for the usefulness of expressing masses in grams, even with reasonable expectations of precison only to nearest 10 g multiples. A mass of 1000 grams is, of course, referred to as a kilogram, with which the children have had some experience by now. The following activities provide expe-riences with masses smaller than a kg and then lead to an extrapolation in the other direction. What is a 1000 kg mass called? What would be measured in tonnes?

activity 1 “i estimate __ grams.” [Meas 4.7/1]

An activity for pairs in which 10 g and 100 g masses are made from Plasticine by balancing them against commercially-made masses. Then the masses of various objects are estimated and measured to the nearest 10 g.

Materials • A sturdy equal-arm, two-pan balance. • A set of commercially-made gram masses. • At least 1 kg of Plasticine. • A collection of objects of various masses in the range 10 g to 1000 g.

What they do 1. A 10-gram mass is placed on one pan of the balance and a ball of Plasticine on the other. Plasticine is added to or removed from the ball until the two masses balance. The procedure is repeated until there are ten 10-g objects.

2. Then, using a similar procedure, 100 g of standard mass(es) are placed on one pan and balanced by a single ball of Plasticine in the other pan. This is repeated until there are 8 more 100-g Plasticine balls.

3. Question: Using all of the standard masses and all of the Plasticine masses, what is the largest mass that can be weighed?

4. The masses of each of the objects in the collection are estimated and then meas-ured (using the balance), recorded, and compared in a table like the one follow-ing. Comparisons are also made with measurements recorded by others.

Object Estimated mass Measured mass grams (g) grams (g)

5. Which of the objects is heaviest? . . . lightest? Can you graph your findings?

discussion of concepts

445

activity 2 “how many grams to a litre? it all depends.” [Meas 4.7/2]

An activity for four providing further practice in estimating, measuring, and ordering mass in grams . . . as well as an experience with relative density.

Materials • A sturdy equal-arm, two-pan balance. • A set of commercially-made gram masses. • The Plasticine masses made in Activity 1. • Two one-litre containers (exact capacity). • Appropriate quantities of materials of various densities, for example: styrofoam

‘peanuts’ (packaging material), puffed rice (or wheat), rice, marbles, beans, water.

What they do 1. Ask each pair to use a chart like that in Activity 1 to record their estimates and then their measures of the mass of one litre of three of the provided materials.

2. They pool their findings, listing the materials in order from from least to greatest mass.

activity 3 a portion of candy [Meas 4.7/3]

An activity for 4 children. Its purpose is to consolidate their new ability to estimate and measure mass in grams.

Materials • Several different kinds of ‘candy’ (whatever candy-sized objects are available). • Small bins or containers for the ‘match and mix’ candies • Small self-sealing plastic bags. • An equal-arm balance and a set of gram masses. • A pack of portion cards on which are written masses in multiples of 5 from 15 g

to 60 g.* * A suitable set is provided in the photomasters

What they do 1. Before beginning, a display of ‘match and mix’ candies is arranged in appropri-ate containers. As a special promotional event in the candy store, the players can win a portion of candy.

2. Players take turns being the checker. The rest are choosers. 3. Each chooser in turn takes the top card from the face-down pile of portion cards. 4. Having taken a card, the chooser selects, by ‘matching and mixing,’ an amount

of candy that she thinks will come close to, but not exceed, the mass on the por-tion card.

5. The checker weighs the candy and informs the chooser of the measured mass. If the amount chosen is within 10 g of the measured mass, the chooser keeps the candies. If not they are returned.

6. After an agreed number of turns, the winner is the player with the most ‘candy.’

Meas 4.7 Grams, tonnes (cont.)

446

activity 4 the largest animal ever [Meas 4.7/4]

An activity for a small group of children working together. Its purpose is to intro-duce the tonne (t) as an appropriate unit when measuring and comparing masses of thousands of kilograms.

Materials • A good encyclopedia.

1. Discuss with the children just how small a gram is and how useful a kilogram is for measuring the masses of many everyday things. How many grams in a kilogram?

2. Does anyone know the name of the standard mass that is 1000 kilograms? (If not, introduce the tonne, sometimes referred to as a ‘metric ton.’)

3. What things do you know about that have a mass of several thousand kilograms? (E.g., whales, elephants, dinosaurs, light rail transit cars, weights and capacities of vehicles, boxcars, ships, . . .) With masses that are so large, the tonne is a use-ful unit.

4. What is the largest animal that you know about? (One encyclopedia entry as-serts that the whale is the largest animal that has ever lived, including prehistoric dinosaurs.)

What they do 1. In small groups, the children make lists of large animals, ordering them from those that they think have the greatest mass to those they think have the least.

2. They check their lists by reference to, say, an encyclopedia, where they will find masses of whales, dinosaurs, and elephants, at least, given in tonnes (also called ‘metric tons,’ especially in U.S. publications). Encourage them to make a chart with the masses in tonnes (t) and in kilograms (kg). [A blue whale, for example, can have a mass of up to 150 t or 150 000 kg.]

3. They discuss their predictions and findings and their understandings of the rela-tive size of a tonne.

4. There might be interest in finding out the weight of a family vehicle and its load-carrying capacity, perhaps comparing minivans with regular cars.

From the tiny gram to the massive tonne (and beyond, in each direction), the metric system provides logically interrelated units suited to a wide range of masses and levels of precision. As with all measurement tasks, the choice of an appro-priate standard unit of measure and a suitable measuring device is central to the measuring process.

OBSERVE AND LISTEN REFLECT DISCUSS

Suggested outline for discussion

Meas 4.7 Grams, tonnes (cont.)

discussion of activities

447

Meas 4.8 RELATIONS BETWEEN STANDARD MEASURES

Concept That of numerical relations between different units of mass.

Ability To convert in both directions between grams, kilograms, and tonnes.

Examples of this concept have already been encountered in the previous topics. In this topic we make the concept explicit and organize particular cases in the form of a table.

activity 1 Completing the table of units of mass [Meas 4.8/1]

An activity for children to do individually. Its purpose is to summarize what they have learned about the relations between units of mass.

Materials • Pencil, paper, and ruler for each child.

What they do 1. Each child copies and completes the table illustrated below. 2. When completed, they compare results and discuss any disagreements. If these

cannot be resolved, they consult their teacher.

TABLE

grams kilograms tonnes g kg t = 1 =

= = 1

1 = =

This is what they should get.

TABLE

grams kilograms tonnes g kg t

1 000 = 1 = .001

1 000 000 = 1 000 = 1

1 = .001 = .000 001

discussion of concept

448

discussion of activity

Meas 4.8 Relations between standard measures (cont.)

Completing the table of units of mass should not only help organize and consoli-date the numerical relationships among them but also can provide a Mode 3 con-text for noticing and reflecting upon the visual non-symmetrical character of the symbols for, say, ‘one thousand’ and ‘one thousandth.’

OBSERVE AND LISTEN REFLECT DISCUSS

449

[Meas 5] Time

MEAS 5.6 LOCATIONS IN TIME: DATES

Concept ‘Where we are’ in time. Abilities (i) Tofindandnamegivendatesinacalendar. (ii) Tofindhowlongatimeitisfromonedatetoanother.

Ifwethinkofourpassageintimeasmovementalongaroad,thedatetellsuswherewearetothenearestday,andthetimeofdaytellsuswithinthatday.Thepresenttopicdealswiththefirstofthese. Acalendarisatableofdaysinwhicheachdayisgivenauniquenamebyspecifyingtheyear,month,anddayofthemonth.Thisnameiscalleditsdate.Inaddition,itkeepstrackofthedaysoftheweek,withinwhichmuchofourdailylifeisplannedforworkandleisure.Anditenablesustoworkouthowlongintimeitisbetweentwogivendates.

Activity 1 What the calendar tells us [Meas 5.6/1]

Ateacher-ledactivityforagroupofanysize.Itspurposeistofamiliarizechildrenwiththeprocessesofnamingandlocatingdatesinthecalendar.

Materials • Asufficiencyofcalendars.* • Iftheprintissmall,theyalsoneedsomethingtopointwith. *Ifpossibleoneperchild,otherwiseonebetweentwo.Foralargegroup,itisuse-

fultohavealsoawallcalendar,largeenoughforalltosee,andapointer.

Suggested 1. Beforetheactivitybegins,havetoday’sdatewrittenontheblackboard.(Many outline for teachersmakeadailypracticeofthis.) the activity 2. Asksomeonetoreadthisaloud,andaskwhatittellsus.(Ittellsuswhichdayof

theweekwearein,whichdayofthemonth,whichmonthoftheyear,andwhichyear.Thatistosay,ittellsuswhereweareintimeandgivesitaname.Thiskindofnameiscalledadate.

3. Tomorrowweshallhavemovedonaday,sowhatwillthedatebethen? 4. Andwhatwasthedateyesterday? 5. Askthechildrentofindtoday’sdateintheircalendars.Iftheyneedhelp,tell

themtobeginbyfindingtherightmonth;Then,thenumberforthedayofthemonth.

6. Pointouthowwhenwehavefoundtherightdateinthemonth,thecalendaralsotellsuswhichdayoftheweekwearein.Sowecancheckoneagainsttheother.

7. Askthemiftheycanfindoutfromtheircalendarswhatthedatewillbeoneweekfromtoday.(Ifthistakesthemintoanothermonth,dostep8beforethisone.)

Discussion of concepts

450

8. Askthemtofindwhatwasthedateaweekbackfromtoday. 9. Repeatfortwo,threeweeksforwardandback.Itbecomesharderwhenthe

monthchanges. 10. Whatdayoftheweekwillitbeonemonthforwardfromtoday?Thatistosay,

onthedaywiththesamenumberastoday,nextmonth.Andonemonthbackfromtoday?

11. Likewiseforothernumbersofmonthsforwardandback. 12. Askiftheyknowthedates(monthandday)oftheirbirthdays.Theycanthen

usetheircalendarstofindwhatdayoftheweekitwillfallonnext. 13. Repeatforotherimportantdates.

Activity 2 “How long is it . . . ?” (Same month) [Meas 5.6/2]

Anactivityforasmallgroupofchildren,whichtheyshouldbeabletodoontheirownafterthepreparationtheyhavehadinActivity1.Itspurposeistogivefurtherpracticeinfindingtheirwayaroundinacalendar.Theymayneedhelpingettingstarted.

Materials • AsforActivity1. • Pencilandpaperforeachchild. What they do 1. Onechildchoosesadatebetweenthepresentdayandtheendofthemonth,and

asks(e.g.)“Howlongisituntilthe27thofthismonth?” 2. Theyallcountforwardinweeks,andthentheextradays.E.g.,if‘today’was

the4thofthemonth,theywouldcount,startingfrom4onthecalendar,“1week”(tothe11th),“2weeks”(tothe18th),“3weeks”(tothe25th),“andtwodays.”

3. Whentheyhaveallwrittendowntheiranswers,theycompareresultsanddis-cussanydisagreements.

4. Afterallhavehadaturn,steps1,2,and3mayberepeatedforadateearlierinthemonth.Inthiscase,thequestionaskedwouldbe“Howlongisitsince...?”

Activity 3 “How long is it . . . ?” (Different month) [Meas 5.6/3]

AnextensionofActivity2.Thestepsareexactlythesame,exceptthatinstep1thestartingchildchoosesadateinthesameyearbutadifferentmonth.E.g.,shemightaskhowlongitisuntilhernextbirthday,ifitisnotyetpast;orsinceherlastbirth-day,ifitis.

Meas 5.6 Locations in time: dates (cont.)

451

Activity 4 “How long is it . . . ?” (Different year) [Meas 5.6/4]

Afurtherextensionoftheprevioustwoactivities.Aneasybeginningwouldbetoask“HowlongisitsincethedayIwasborn?”Theyalreadyknowhowmanyyearsthisis,soitisonlyamatterofcountingforwardinmonthsanddayssincetheirlastbirthdays.Otherpossibilitiesarehistoricalevents,preferablyoneswhichhavesomeinterestforthechildren.

Computerprogramsareavailablewhichwillprintacalendarforanyrequiredyear,past,presentorfuture,anditaddstotheinterestifthesecanbemadeavailableforthisactivity.Theycanthenfindonwhichdayoftheweektheeventtookplace,aswellashowlongagoitwas.

Futureaswellaspasteventsshouldofcoursebeincluded.Itmaybeusefultohavesomesuggestionsreadyincasetheyareshortofideas.

Thecalendarisanexcellentrepresentationofthecombinationofthreeseparateorderingsbywhichweorganizeourexperienceofthepassageofdays,andgivenamestothem.Thesehavealreadybeendealtwithseparatelyinprevioustop-icsontimeinSAIL Volume 1,andthepresenttopicbringsthemtogether.Givingthemactivitieswhichinvolverepeateduseofthecalendarshouldhelpconsolidatechildren’sknowledgeoftherelationshipsbetweendaysoftheweek,daysofthemonth,andmonthsoftheyear,anddeveloptheirabilitytoputthisknowledgetouse.

Discussion of activities

Meas 5.6 Locations in time: dates (cont.)

OBSERVE AND LISTEN REFLECT DISCUSS

“How long is it . . . ?” (Different year) [Meas 5.6/4]

452

Meas 5.7 LOCATIONS IN TIME: TIMES OF DAY

Concepts (i) Whereweareduringaday. (ii) Theclockasaninstrumentformeasuringthetimeofday.

Abilities Totellthetimeofdaybytheclock,inbothdigitalandanaloguenotations.

Theprecedingtopicwasconcernedwithlocationintimetothenearestday.Inthepresenttopic,wecontinuethedevelopmentofthisconcepttospecifylocationswithinthetimestretchofasingleday.Theclockreplacesthecalendarasthemeanswherebywedothis,andthesymbolismbywhichwedescribeandrecordit.

Activity 1 “How do we know when to . . . ?” [Meas 5.7/1]

Ateacher-leddiscussionforagroupofanysize.Itspurposeistounderlinetheim-portanceofbeingabletotellthetime,asawayoffittingone’sownactivitiesinwiththoseofothers.

Materials Nonerequired.

Suggested 1. Ask“Howdoweknowwhento...?”Thiscanbeanythingwhichmustbe outline for doneataspecifictime,suchascatchthebusforschool,switchonthe the discussion televisionforafavouriteprogram,comehomefromvisitingafriend.Insome

cases,e.g.,endthelessonandgooutforrecess,theanswermaywellbethatsomeoneelsesaysso:e.g.,mostchildrenwillbetoldbyaparentwhenitistimetogetupinthemorning.Inthiscase,wewanttoknowhowtheyknow.Ifbellsringatprogrammedtimesinyourschool,youneedtobepreparedfortheanswer“Becausethebellrings.”

2. Innearlyeverycase,theanswereventuallycomesbacktobecauseitisapar-ticulartimeofday.ItwasinordernottoimplythisanswerinthewordingofthequestionthatIdidnotnamethisactivity“Howdoweknowwhenit’stimeto...?”

3. “Sowhendoweknowwhenitis...(whatevertimeisgiven)?” 4. Theanswerwillnearlyalwaysbe“Bylookingataclockorwatch.” 5. Soweneedtobeabletoreadthetimefromaclockorwatch,ifwecannot

already.

Discussion of concepts

453

Activity 2 “Quelle heure est-il?” [Meas 5.7/2]

Ateacher-leddiscussionforagroupofanysize.Itspurposeistointroducechildrentotheanalogueclockface.

Materials • Ananalogueclockface,withamovablehourhandonly.Thisneedstobelargeenoughforthewholegrouptoseeeasily.

Suggested 1. Explainthatlongago,peopleonlyneededtotellthetimetothenearesthour,so outline for whentheywantedtoknowthetimetheyusedtoask“Whathourisit?”In the discussion Frenchthisishowtheystillaskwhatthetimeis. 2. Sotheearlyclockfacesonlyshowedhours,andthisishowwearegoingto

begin. 3. Earlierstill,beforetherewereanyclocks,therewasonetimeofdaywhichthey

couldtellbythesun(ifitwasshining).Cananyonesaywhatthismightbe? 4. Ifanyonesuggestssunriseorsunset,thesearegoodideas,butwhywon’tthey

do?(Thesechangefromdaytoday.) 5. Whatdoesn’tchangeeventhoughthedaysgetshorterorlongeristhemiddleof

thedaywhenthesunishalfwaybetweenrisingandsetting.Thisiswhenitishighestinitspathacrossthesky.(Later,youmightintroducetheword‘zenith.’)

6. Sotheydividedthedayintotwoequalparts,beforemiddayandaftermidday.Whenwriting,theseareoftenwritten‘a.m.’and‘p.m.’forshort.‘A.m.’standsfor‘ante meridiem,’whichisLatinfor‘beforemidday.’‘P.m.’standsfor‘post meridiem,’whichmeans...?Whichisitnow,a.m.orp.m.?

7. Therearetwelvehoursineachhalfday.Thea.m.halfstartsat...?Andendsat...?Andthep.m.partstartsat...?Andendsat...?

8. Nowshowtheclockface,withthehandpointingtotheactualtime,tothenear-esthour.Ask“Quelleheureest-il?”,or“Whathourisit?”,whicheveryoupre-fer.Sinceweareimaginingourselvestobeinthepositionofpeoplelongago,theymightliketoanswerastheydid,andsay(e.g.)“Itiselevenoftheclock.”Laterthiswillbeshortenedto“o’clock.”

9. Thenextquestionishowweknowwhetherthehourshownontheclockfaceisa.m.orp.m.Thisisusuallyclearfromwhereweareandwhatwearedoing.Intheforegoingexample,wherewouldtheybeifitwaselevenp.m.?Wehope,inbedandasleep!Ifitisclearwhichwemean,wedonotneedtosaya.m.orp.m.unlesswewantto.

10. Theyarethenreadytopractisereadingthehourfromtheclockface,withthehourhandalwayspointingtoanexacthour.Itshouldnottakelongforthemtobecomefluentatthis,afterwhichtheywillbereadyforthenextactivity.

Meas 5.7 Locations in time: times of day (cont.)

454

Activity 3 Hours, halves, and quarters [Meas 5.7/3]

AcontinuationofActivity1,whichusestheminutehandtoshowhalvesandquartersofanhour.

Materials • ThesameclockfaceasusedforActivity2. • Asecondone,similarbutwithaminutehandaswellasanhourhand. 1. Usethefirstclockfacetoexplainthatthehourhandmovesslowlyroundthe dial,takinganhourtogetfromonefiguretothenext. 2. Positionthehourhandhalfwaybetweentwohours,andaskwhattheywouldcall

thistime. 3. Acceptanysensibleanswers.Nowshowthesecondclockface,andexplainthat

thishelpsustotellthetimemoreexactly.Thelongerhandmakesacompleteturnineveryhour.

4. Starttheminutehandat12,andask:“Startinghere,whatpartofaturnhasitmadewhenitgetstohere?”,movingittothefigure6.(Halfaturn.)“Sohowlongafterthehourdoesitshow?”(Halfanhour.)

5. Putthehourhandhalfwaybetweenanytwofigures,say,9and10,withtheminutehandat6,andaskwhattimethisisshowing.“Halfanhourafternine”isagoodanswer,whichmaybeshortenedto“Halfpastnine.”

6. Movethehourhandtohalfwaybetweentwootherfigures,andask“Whattimedoestheclockshownow?”Repeatuntiltheyarefluent,whichshouldnottakelong.

7. Repeatsteps4,5,and6,asbefore,exceptthattheminutehandnowshowsaquarterafterthehour.

8. Repeatsteps4,5,and6,asbefore,exceptthattheminutehandnowshowsaquarterbeforethenexthour.Though“Aquarterbefore...”istheusualanswer,Ithinkthatweshouldalsoaccept“Threequartersafter...”,first,becauseitisalsocorrect,and,second,becauseitcorrespondstothewayitwouldbebothspokenandwritteninhoursandminutes,e.g.,9:45.

9. Finally,repeattheabovewiththeminutehandat12.Explainthatwhenthetimeisrightonthehour,wereadthisas(e.g.)“Threeo’clock.”

10. Onasubsequentoccasion,thechildrenmayusefullypractisereadingtheclockfaceasabove,withtheminutehandinoneofthefourpositionsabove.Theymaytakeitinturnstosetthehandswhiletheothersreadthetime.

Activity 4 Time, place, occupation [Num 5.7/4]

Anactivityforasmallgroupofchildren.Itspurposesaretogivefurtherpracticeinreadingthetimefromaclockface,andtorelatetimesofdaytoawidercontextofwheretheygoandwhattheydo.

Materials • Aclockface,withmovablehourandminutehands. • Atwo-sidedcard,with‘a.m.’ononesideand‘p.m.’ontheother.

Meas 5.7 Locations in time: times of day (cont.)

Suggested outline for the discussion

455

What they do 1. Onechildhastheclockface,andsetsittoatimeinwhichtheminutehandisatoneofthefourquartersandthehourhandisinapproximatelytherightpositionrelativetotheminutehand.Thetwosidedcardmaybeeithersideup,butforthefirstroundthetimeshouldbeearlierthanthepresenttime.

2. Hefirstaskstheothers“Whattimeisthis?” 3. Whentheyhaveanswered,hethenaskstheotherchildreninturn,“Wherewere

youatthistime(pointingtotheclock)today?Andwhatwereyoudoing?” 4. Whenallhaveanswered,theturnpassestothenextchildwhorepeatssteps1,

2,and3,asbefore,exceptthatthetimeshownislaterthanthepresenttime,andthequestionsinstep3arechangedto“Wherewillyouprobablybeatthis time (pointingtotheclock)today?Andwhatwillyouprobablybedoing?”

5. Thenextchildsetsthetimetoanydesiredtimeofday,thequestionsinstep3beingchangedto“Wherewereyouatthistime(pointingtotheclock)yester-day?Andwhatwereyoudoing?”

6. Thenextchilddoeslikewise,butasks“Wherewillyouprobablybeatthis time (pointingtotheclock)tomorrow?Andwhatwillyouprobablybedoing?”

7. Fourchildrenwillnowhaveactedasquestioner.Ifthereareotherswhohavenotyethadaturnatthis,theymaymakeanysensiblechoiceofdayfortheirquestions.

Theobjectofallthisisnotonlyforthemtobeabletotellthetimebytheclock,buttounderstandthiswithinmeaningfulcontexts.HereIsuggesttwo.Firstisthepracticalneedofanagreedsystemfordescribinglocationsintime,forsocialcoop-eration.Manyofthewaysinwhichwecoordinateourownactionswiththoseofotherswouldfalltopieceswithoutclocksandwatches.Becauseofthispracticalneed,theoriginanddevelopmentoftimemeasurementhasalonghistory.Ihavetriedtoshowjustaglimpseofthis. Bothanalogueanddigitalclockfacesarenowwidespread,andsometeach-ersprefertoteachthelatterfirstonthegroundsthatitiseasier.Thismaybeso,thoughitbeginswithhoursandminutesratherthanhoursandquarters.Myself,Iprefertostartwiththeanalogueclockface,partlyforthehistoricalbackgroundwhichIhavealreadymentioned,andpartlybecauseforsomepurposesIfinditpreferable.Theanalogueclockfaceshowsintervalsgraphically,sothatitiseasytoreadoffhowlongitis(e.g.)from9:45to10:20.At,say,06:55(digital)thepo-sitionoftheminutehandat5minutesbeforethehourshowsdirectlythatthetimeiscominguptoseveno’clock.Thedigitalreadingrequiresthistobeinferredfromthedifferencebetween55and60.Finally,thedigitalreadingsgiveagreaterde-greeofaccuracythanisneededformanypurposes.WhenIlookatmyownwatch,ittellsmethetimetothenearestminuteandsecond,whetherIwanttheseornot.(Yes,itisdigital,becauseIliketheotherfacilitieswhichcomewiththis.)ButinourlivingroomwehaveananalogueclocktoanoldGermandesign. Childrenneedtobeathomewithbothsymbolisms,thespatialandthenumeri-cal.Above,IhaveexplainedwhyIhavebegunwiththeformer,butifyouprefertheotherwayaround,theforegoingactivitiescanbetakeninadifferentorder.

Discussion of activities

Meas 5.7 Locations in time: times of day (cont.)

OBSERVE AND LISTEN REFLECT DISCUSS

456

Meas 5.8 EqUIVALENT MEASURES OF TIME

Concept Thatthesamedurationoftimecanbeexpressedindifferentunits.

Ability Torecognizeequivalentmeasuresoftime,i.e.,thesamedurationexpressedindifferentunits.

Equivalencebetweenasinglelargerunitandaparticularnumberofsmallerunits(e.g.,1hourisequivalentto60seconds)hasalreadybeenintroduced,in‘Timesheets,’Meas5.5/1inVolume 1.Hereweexpandtheuseoftheconcept,torecog-nizeequivalencessuchashourand45minutes.Ihavekepttoinstanceswhicharelikelytobemetwithineverydaylife.

Activity 1 Pairing equivalent times [Meas 5.8/1]

Agameforasmallgroupofchildren.Itspurposeistointroduceandpractisethemorefrequentlyusedequivalencesbetweendays,hours,minutes,andseconds.

Materials • Apackofcardsshowingpairsofequivalenttimes(durations).* *Photomastersareprovidedforapackcontaining15pairs.

What they do 1. Thecardsareshuffledandputinapilefacedownonthetable.Thetopcardisturnedfaceupandputseparately.

2. Theplayersthentaketurnstoturnoverthetopcardfromtheface-downpile,andputitfaceupwith(notontopof)theothers.

3. Iftheplayerwhoseturnitisseestwocardswhichshowthesamedurationoftimeindifferentunits,sheclaimsthem.

4. Asthenumberofface-upcardsincreases,clearlythechanceoftherebeingtwoequivalentmeasureincreases.

5. Iftheplayerwhoseturnitisoverlooksapair,thenextplayermayclaimitbeforetakingherturninthenormalway.Andifthisplayeroverlooksit,thesameap-pliestothenextplayerandsoon.

6. Thewinneristheplayerwithmostpairswhenallhaveplayedouttheirhands.

Thisisastraightforwardactivity.However,itdoesencouragementalconversionofeachnewcardasitisturnedover,inordertolookforanequivalenttimeamongthosealreadyshowing.

OBSERVE AND LISTEN REFLECT DISCUSS

Discussion of concept

Discussion of activities

34

457

[Meas 6] TemperaTure

MEAS 6.2 MEASURING TEMPERATURE BY USING A THERMOMETER

Concept The thermometer as an instrument for measuring temperature.

Abilities (i) To use a thermometer correctly for measuring temperature. (ii) To sequence two or three objects in order of temperature.

Children will already be familiar with the idea of measuring instruments, such as rulers for measuring length, kitchen scales, speedometers in cars. They will prob-ably also have encountered a thermometer in some form or other — most children have had their temperatures taken by a parent or doctor. So here again, we are consolidating, organizing, and extending their everyday knowledge.

activity 1 The need for a way of measuring temperature [Meas 6.2/1]

A teacher-led experiment followed by a discussion. Its purpose is to underline the need for a way of measuring temperature which is more accurate than estimation, and to think of other reasons for this. Below I suggest two forms of the experiment. You may like to try others.

Materials • Four containers for water, such as mugs, jam jars, . . . .

Suggested 1. Thefourcontainersarenearlyfilledwithwateratthreedifferenttemperatures. procedure One should contain cold water, chilled if necessary with a lump of ice which for the should be removed before the experiment. We may call this one C. Another, experiment container H, should contain hot water, as hot as would be comfortable to wash

one’s hands in. The remaining two should contain warm water, both at exactly the same temperature. This should be just a little above room temperature. We will call these W (for warm). Note that these terms are for our own conven-ience: the containers should not be labelled with these letters.

2. First put out containers C and H. 3. Avolunteerisnowneeded.AskhimtoputtwofingersofhislefthandinC and

twofingersofhisrighthandinH, and say which he thinks is hotter and which is colder.Heshouldkeephisfingersinthecontainersfornotlessthan20secondsbefore the next step.

4. Now give him the two W containers and ask him to compare these. (In this case it is better to say “compare” rather than ask which is hotter/colder.)

5. Although they are in fact at the same temperature, one will feel warmer to the fingerscomingfromthecoldwater.

6. Now ask another volunteer to compare the two W containers. He is likely to say that they are both the same.

Discussion of concept

458

Meas 6.2 Measuring temperature by using a thermometer (cont.) 7. So here we have one reason why we need a way of measuring temperature more

reliably than by feeling. Can anyone think of others? (E.g., too hot or too cold to be safe to touch; inaccessibility; greater accuracy required than would be pos-sible by estimation.)

Variation In this simpler form of the experiment, only one W container is used. After step 3,thevolunteerputsthefingersofbothhandsinW, which will feel warmer to one hand than to the other. Since this does not take long, more children can take part. May I suggest that in any case you try this for yourself beforehand? It is a little strange to have ones hands sending messages which are contrary to what one knows to be the case.

activity 2 using a thermometer [Meas 6.2/2]

An activity for a small group. After a teacher-led introduction, they should be able to continue on their own. Its purpose is to introduce children to the use of a thermometer.

Materials • TwoCelsiusthermometers,scaledfrom0°(orbelow)to100°(orabove). • At least two blocks of different heights. The more of these that can be made

available, the better. • A number of containers for water, such as mugs, jam jars, . . . • Hot and cold tap-water. • Pencil and scrap paper for each child.

Suggested 1. Fill two of the containers with water at clearly different temperatures. introduction 2. Have the children agree which is the hotter and which the colder, and let them

show this as before by putting them on blocks of different heights. 3. Now put a thermometer in each of the containers, and ask the children to watch

carefully and report what happens. 4. We hope that at least the nearer ones will observe that the thread of mercury or

alcohol changes length, doing this more slowly until it is steady. 5. Put the thermometers vertically side by side, in the same relative positions. We

hope that they will now notice that the greater height in the thermometer corre-sponds to the higher temperature.

6. Looking more closely, they can see that the thermometer is marked with a scale. These are degrees Celsius, which are the international units used to measure temperature.

7. Pass them around, and let each say what the reading is. Unless the liquid was at room temperature, the reading will gradually change. What do they learn from this?

8. Theymaynoweachtakeacontainerandfillitwithwatermixedfromcoldandhot taps. Working in pairs, one estimates which is hotter and which is colder, and shows this by putting them on blocks.

459

9. They then check with thermometers. Since there will certainly not be two of these per pair of children, they will have to make the comparison by reading the temperatures in degrees Celsius which they record on a small piece of scrap paper.

10. Thesearewettedsothattheyadheretemporarilytothecontainers. 11. (Optional further step.) Pairs may then cooperate to put all their containers in

order of temperature.

Note The temporary nature of the labels’ attachment corresponds well to the fact that the temperatures will not stay the same for long!

Note that in Activity 2, the children begin by using their own senses to compare temperatures, so that the higher thermometer reading corresponds to what they already know to be the higher temperature. This relates the new experience to their existing schemas. I have not included any explanation of how a thermometer works, mainly be-cause these are resources for learning and applying mathematics rather than an introduction to science. It will, however, be as well to be prepared with an expla-nation if children ask. The most common kind of thermometer, and the one which you will need for use in the classroom, is the mercury-in-glass type, with alcohol thermometers available for sub-zero temperatures. Non-liquid thermometers, which use a bimetallic strip, are also fairly common, so you might want to check out this kind too.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 6.2 Measuring temperature by using a thermometer (cont.)

Discussion of activities

Using a thermometer [Meas 6.2/2]

460

MEAS 6.3 TEMPERATURE IN RELATION TO EXPERIENCE

Concept The relationship between temperatures as shown by a thermometer and everyday

experience.

Abilities (i) To say roughly what temperatures are to be expected in a variety of everyday examples.

(ii) Conversely, to say whether a given temperature is a likely one in any given case.

The concept itself is the same as the children learned in Meas 6.2. Here they are expandingitsfieldofapplication.

activity 1 everyday temperatures [Meas 6.3/1]

A project-like activity for a small group of children to work at cooperatively, as de-scribed below.

Materials • Not less than two thermometers for the group; more if possible. • Pencil and paper for each child.

What they do 1. Explain that now they know how to measure temperature accurately with a ther-mometer, you would like them to make a list of everyday temperatures which they could measure.

2. They should start by ‘brainstorming’ to produce a list of what they want to meas-ure. Here are some suggestions to start with:

Inside temperatures, in the classroom and elsewhere in the school. Outside temperatures, at different times of year. Temperaturesinthesameroomatfloorlevelandasneartheceilingaspracti-

cal. Temperatures of liquids left in shade and sun. Temperatures of ice water, and (under supervision) boiling water. Body temperature using ordinary and (if available) a clinical thermometer.

Why is body temperature important? 3. They work in pairs to read and record the temperatures they have listed, having

firstapportionedthisamongthemselves. 4. Finally they combine their information in a list for the classroom wall.

Discussion of concept

461

activity 2 “What temperature would you expect?” [Meas 6.3/2]

An activity for any sized group of children, working in pairs. Its purpose is to give practice in using and extrapolating their knowledge of likely temperatures, based on the experience gained in Activity 1.

Materials For each pair: • The query list, see below.* • Pencil and paper. • Their rough drafts of the lists made in Activity 1 would be useful, as an aid to

memory. * Note This is provided for your convenience, here and in the photomasters, but you

may wish to extend this to include material of local and topical interest.

What they do 1. They discuss with their partners what would be reasonable answers to the que-riesonthelist,andfilltheseinontheirsheet.Inmostcasesarangeoftempera-tures, or “Around . . .” , would be sensible replies.

2. All the pairs at the same table may then share their conclusions, and discuss any points of divergence. If these cannot be resolved, they will need to consult you.

3. AsinActivity1,thefinalresultsmightmakeasuitabledisplayfortheclassroomwall.

Query List

Whattemperaturewouldyouexpecttofindinthefollowing? A classroom or living room. A refrigerator. A freezer. An oven ready for cooking. A shopping mall. Water from a cold tap. Water from a hot tap. A swimming pool. A hot bath or shower. A mountain stream.

Meas 6.3 Temperature in relation to experience (cont.)

462

activity 3 Temperature in our experience [Meas 6.3/3]

This is a long-term project, for the whole class, to be organized in whatever way you think best. It is based on the idea that temperature is an interesting source of data for environmental studies.

Materials • Suitable display material for the classroom wall, to incorporate the data below as it is collected.

• Celsius thermometers. • An outdoor thermometer, which can be read through the window, is an interest-

ing source of data. A maximum and minimum thermometer would be even more interesting, and could be used to make a graph of year-long changes.

What they do (These are suggestions offered as a starting point.) Temperatures are recorded and collated, in some cases graphically, for the following

temperatures: Outdoors (locally) in summer, winter, spring, fall, at the same times of day. Highest temperature recorded, and lowest. Hourly readings inside and outside for the whole of a school day.

InMeas6.1webasedthescientificconceptoftemperatureonchildren’severydayexperience of this. In Meas 6.2, we introduced the instrument (thermometer) and units (degrees Celsius) by which this is measured. In the present topic, children are developing connections in the reverse direction, from temperature as meas-ured with a thermometer back to everyday experience. This continues the overall process by which we accept the importance and validity of the knowledge which children already have, lead them to consolidate and organize it, and then help them to expand it further.

OBSERVE AND LISTEN REFLECT DISCUSS

Meas 6.3 Temperature in relation to experience (cont.)

Discussion of activities

Temperature in our experience [Meas 6.3/3]

463

Glossary

These are words which may be unfamiliar, or which are used with special-ized meanings. The definitions and short explanations given here are intended mainly as reminders for words already encountered in the text, where they are discussed more fully. This is not the best place to meet a word for the first time.

abstract (verb) To perceive something in common among a diversity of experiences. (adj.) Resulting from this process, and thus more general, but also more remote from direct experience.

add This can mean either a physical action, or a mathematical operation. Here we use it with only the second meaning, in order to keep these two ideas distinct.

addend That which is to be added.base The number used for grouping objects, and then making groups of these groups,

and so on. This is a way of organizing large collections of objects to make them easier to count, and is also important for place-value notation.

binary Describes an operation with two operands.canonical form When there are several ways of writing the same mathematical idea, one of

these is often accepted as the one which is most generally useful. This is called the canonical form. A well known example is a fraction in its lowest terms.

Property which is the basis for classification, and for membership of a given set.commutative This describes a physical action or a mathematical operation for which the result

is still the same if we do it the other way about. E.g., addition is commutative, since 7 + 3 gives the same result as 3 + 7; but subtraction is not.

concept An idea which represents what a variety of different experiences have in com-mon. It is the result of abstracting.

congruent Two figures are congruent if one of them, put on top of the other, would coin-cide with it exactly. The term still applies if one figure would first have to be turned over.

contributor One of the experiences from which a concept is abstracted.counting numbers The number of objects in a set. The cardinal numbers, 1, 2, 3, . . . (continuing

indefinitely). Zero is usually included among these, but not negative or frac-tional numbers.

digit A single figure. E.g., 0, 1, 2, 3, . . . 9.equivalent Of the same value.extrapolate To expand a schema by perceiving a pattern and extending it to new applica-

tions. A concept which is itself abstracted from other concepts. E.g., the concept of an even number is abstracted from numbers like 2, 4, 6, . . . so even

number is a higher order concept than 2, or 4, or 6, . . .interiority The detail within a concept.interpolate To increase what is within a schema by perceiving a pattern and extending it

inwards.

characteristic property

higher order concept

464

low-noise example An example of a concept which has a minimum of irrelevant qualities. The opposite of higher order concept, q.v.

match To be alike in some way. See operation.

Mode 1 Schema building by physical experience, and testing by seeing whether predic-tions are confirmed.

Mode 2 Schema building by receiving communication, and testing by discussion.Mode 3 Schema construction by mental creativity, and testing whether the new ideas

thus obtained are consistent with what is already known.model A simplified representation of something. A model may be physical or mental,

but here we are concerned mainly with mathematical models, which are an im-portant kind of mental model.

natural numbers The same as the counting numbers.notation A way of writing something.numeral A symbol for a number. Not to be confused with the number itself.oblong A plane figure having four sides and four right angles, with adjacent sides un-

equal. An oblong and a square are two kinds of rectangle, in the same way as boys and girls are two kinds of children.

operand Whatever is acted on, physically or mentally.operation Used here to mean mental action, in contrast to a physical action.pair A set of two. Often used for a set made by taking one object from each of two

existing sets: e.g., a knife and a fork. A way of writing numbers in which the meaning of each digit depends both on

the digit itself, and also on which place it is in, reading from right to left.predict To say what we think will happen, by inference from a suitable mental model.

Not the same as guessing. Prediction is based on knowledge, guessing on igno-rance.

schema A conceptual structure. A connected group of ideas.set A collection of objects (these may be mental objects) which belong together in

some way.subitize To perceive the number of objects in a set without counting.sum The result of an addition.symmetry A relation of a figure with itself, in which there is an exact correspondence of

size and shape on opposite sides of a line or around a point. Both line symmetry and rotational symmetry are considered in this book.

transitive A property of a relationship. E.g., if Alan is taller than Brenda, and Brenda is taller than Charles, then we know also that Alan is taller than Charles. So the relationship ‘is taller than’ is transitive.

unary Describes an operation with a single operand.

lower order concept

mathematical operation

place-value notation

465

AlphAbeticAl list of ActivitiesA computer-controlled train 399A new use for the multiplication square 204A portion of candy 445A race through a maze 361Abstracting number sentences 152Advantages and disadvantages 406Air freight 119Airliner 146Alias prime 76, 198Alike because . . . and different because . . . 384“All make an angle like mine.” 270“All put your rods parallel/perpendicular to the

big rod.” 264An odd property of square numbers 79“And what else is this?” 294Angle dominoes 274Angles in the environment 272Animals through the looking glass 314Animals two by two 314“Are calculators clever?” 258Area of a circle 421Area of a parallelogram 417Area of a triangle 419

Big Giant and Little Giant 156Big numbers 102, 103Binary multiplication 159Blind picture puzzle 393Building sets of products 162Buyer, beware 398Buying grass seed for the children’s garden 424Buying smallholdings 426

Calculating in hectares 426Calculating in square metres 423Calculating lengths from similarities 285“Can we subtract?” 138Can you spot a litre? 434Can you think of . . . ? 366Candy store: selling and stocktaking 148“Can’t cross, will fit, must cross.” 275Cards on the table 166Cargo Airships 214

Cargo boats 172Carpeting with remnants 412Cashier giving fewest coins 95Catalogue shopping 124Centre and angle of rotation 352Chairs in a row 396Change by counting on 126Change by exchange 125Checking a spring balance 443Circles and polygons 299Circles and their parts in the environment 267Circles in the environment 266Claim and explain 412Claim and explain (harder) 415Claiming and naming 248Classifying polygons 278Classifying quadrilaterals 292Classifying triangles 287Collecting symmetries 318Colouring pictures 264Combining the number sentences 190Compass directions 332Completing the product table 164Completing the table 428Completing the table of units of capacity or

volume 436Completing the table of units of mass 447Congruent and similar polygons 283Congruent and similar triangles 289Constructing rectangular numbers 74, 194Constructing the results of slides 329Continuing the pattern 72Counting centimetres with a ruler 386Cycle camping 121

Decorating the classroom 389Different name, same angle 339Different objects, same pattern 381Different questions, same answer. Why? 187Directions and angles 338Directions for words 333Drawing mirror images 315Drawing nets of geometric solids 309Drawing the number line 359

466

Equivalent fraction diagrams (decimal) 242Equivalent measures, cm and mm 397Escape to freedom 341Everyday temperatures 460Expanding the diagram 233Explaining the shorthand 178

Factors bingo 197Factors rummy 197Feeding the animals 227Fractional number targets 262Fractions for sharing 255Front window, rear window 141Front window, rear window – make your own

143

Geometric nets in the supermarket and elsewhere 312

Get back safely 345Gift shop 136Gold rush 409

“Hard to know until we measure” 408“Hard to tell without measuring” 433Head keepers 230‘Home improvement’ in a doll’s house 414Honest Hetty and Friendly Fred 439Hopping backwards 360Hours, halves, and quarters 454“How are these related?” 83“How can we write this number?” (Headed

columns) 373“How do we know that our method is still

correct?” 252“How do we know when to . . . ?” 452“How long is it . . . ?” (Different month) 450“How long is it . . . ?” (Different year) 451“How long is it . . . ?” (Same month) 450How long will the frieze be? 391“How many cubes in this brick?” (Alternative

paths) 179“How many grams to a litre? It all depends.”

445“How would you like it?” 97

“I can see . . . ” 301“I estimate __ grams.” 444“I know a shortcut.” 411“I know another way.” 163“I think you mean . . .” 294“I’ll take over your remainder.” 210“I’m thinking in hundreds . . . .” 209Instant tiling 407Introducing the decimal point 374Introduction to back bearings 344Introduction to rotational symmetry 355Introduction to symmetry 317Inventing tessellations 306Is there a limit? 366Island cruising 334“It has to be this one.” 429

Largest angle takes all 271Less than, greater than 100Little Giant explains why 158Long multiplication 182

Mailing parcels 442Mailing parcels (spring balance) 443Making a set of kilogram masses 441Making and tasting (accuracy in the kitchen) 435Making equal parts 219Making jewellery to order 240Match and mix (line symmetry) 357Match and mix: equivalent decimal fractions 244Match and mix: equivalent fractions 239Match and mix: parts 225Match and mix: polygons 279Match and mix: rotational symmetries 356Match and mix: triangles 288Measuring by counting tiles 406“Mine is the different kind.” 275Mixed units 431Model bridges 390Mountain road 387Mr. Taylor’s Game 191Multiples Rummy 168Multiplying 3-digit numbers 172Multiplying by 10 or 100 176Multiplying by hundreds and thousands 178Multiplying by n-ty and any hundred 180“My rods are parallel/perpendicular” 263

467

Naming big numbers 104Net of a box 415Number stories (multiplication) 151Number stories, and predicting from number

sentences 153Number targets 86Number targets beyond 100 87Number targets by calculator 259Number targets in the teens 89Number targets using place-value notation 91Number targets: division by calculator 217

Odd sums for odd jobs 116One tonne van drivers 122

Pair, and explain 243Pairing equivalent times 456Parallels by sliding 330 Parcels within parcels 201Parts and bits 223Parts of a circle 266Patterns in sound 382Patterns which match 382Patterns with circles 268Place-value bingo 91Planning our purchases 118“Please be more exact” (Telephone shopping)

392“Please may I have?” (Diagrams and notation)

235“Please may I have?” (Metre and related

units) 403Pointing and writing 375Polygon dominoes 279Predict, then press 257Problem: to put these objects in order of mass

438Products practice 166Putting and taking 71Putting containers in order of capacity 433

Q and R ladders 213“Quelle heure est-il?” 453Quotients and Remainders 205

Race from 500 to 0 144Race to a litre 436Reading headed columns in two ways 245Recognizing solids from their nets 311Relating different units 402Relations between quadrilaterals 293Renovating a house 117Renting exhibition floor space 424Right angles, acute angles, obtuse angles 273Rounding big numbers 111Rounding decimal fractions 261Rounding to the nearest hundred or thousand

109Run for shelter 107

Same kind, different shapes 221Same number, or different? 247“Same number, or different?” 99Seeing, speaking, writing 11-19 88Sequences on the number line 360Shapes and sizes 408Shrinking and growing 376Sides of similar polygons 284Similarities and differences between patterns

383Sliding home in Flatland 326Snail race 370Snails and frogs 371Sorting equivalent fractions 238Sorting parts 223Sorting proverbs 72“Special: Small Lemonade, 10¢” 435Square numbers 79Start, Action, Result beyond 100 121Start, Action, Result up to 99 113Subtracting from teens: “Check!” 135Subtracting from teens: choose your method

133Subtracting three-digit numbers 145Subtracting two-digit numbers 139Sum of two primes 77

Taking 360Talk like a Mathematician (lines, rays,

segments) 322

468

Target, 1 251“Tell us something new.” 82Temperature in our experience 462Tens and hundreds of cubes 67Tens and hundreds of milk straws 68Tessellating any quadrilateral 307Tessellating other shapes 305Tessellating regular polygons 304“That is too exact.” (Car rental) 395“That is too exact.” (Power lines) 394The largest animal ever 446The need for a way of measuring temperature

457The need for standard units 385The rectangular numbers game 75, 195The sieve of Eratosthenes 77, 199Thousands 68Throwing for a target 69Tiling the floors in a home 416Till receipts 126Till receipts up to 20¢ 135Time, place, occupation 454Trainee keepers, qualified keepers 229Treasure chest 183Triangle dominoes 288Triangles and larger shapes 302Triangles and polygons 298True north and magnetic north 350True or false? 323

Unpacking the parcel (binary multiplication) Alternative notations 159

Unpacking the parcel (division) 190Unpacking the parcel (subtraction) 131Using a thermometer 458Using multiplication facts for larger numbers

171Using set diagrams for comparison 129Using set diagrams for finding complements

130Using set diagrams for giving change 130Using set diagrams for taking away 128Using the short lengths (Power lines) 395

Village Post Office 207

Walking the planks 353Walking to school 320“We don’t need headings any more.” 90“What could stand inside this?” 428What must it have to be . . . ? 296What number is this? (Double starter) 365What number is this? (Single starter) 363What number is this? (Decimal fractions) 369“What temperature would you expect?” 461What the calendar tells us 449Where are we? 347Where must the frog land? 360“Which angle is bigger?” 271“Will it, won’t it?” 405“Will this do instead?” 237Words from compass bearings 339

469

Sequencing guideS

grade Levels 3 through 6

470

471

A Sequencing guide for grade Level 3

Review Patt 1.4/1, 2, 3, 4, 5 (pages 381-384)Review Num 1.9 and 1.10 in SAIL Volume 1 Num 1.11 Extrapolation of number concepts to 100 /1 Throwing for a target 69

/2 Putting and taking 71Num 1.12 Ordinal numbers, first to one hundredth /1 Continuing the pattern 72

/2 Sorting proverbs 72Org 1.13 Base ten /1 Tens and hundreds of cubes 67 /2 Tens and hundreds of milk straws 68 /3 Thousands 68Num 2.8 Written numerals 20 to 99, headed columns /1 Number targets 86 /2 Number targets beyond 100 87Num 2.10 Place-value notation /1 “We don’t need headings any more.” 90 /2 Number targets using place-value notation 91

/3 Place-value bingo 91Num 2.11 Canonical form /1 Cashier giving fewest coins (stage a) 95 /2 “How would you like it?” 97Num 2.12 The effects of zero /1 “Same number, or different?” (stage a) 99

/2 Less then, greater than (stage a) 100Num 2.13 Numerals beyond 100, written and spoken /1 Big numbers (to thousands) 102

/2 Naming big numbers (to thousands) 104Return to: Num 2.11/1, stage b), Num 2.12/1, 2, stage b) Review Num 3.7 and 3.8 in SAIL Volume 1 Num 3.9 Adding, results up to 99 /1 Start, Action, Result up to 99 113 /2 Odd sums for odd jobs 116

/3 Renovating a house 117 /4 Planning our purchases 118

/5 Air freight 119Num 3.10 Adding, results beyond 100 /1 Start, Action, Result beyond 100 121 /2 Cycle camping 121

/3 One tonne van drivers 122 /4 Catalogue shopping 124

Review Num 4.6, 4.7, and 4.8 as necessaryNum 4.9 Subtraction up to 99

/1 “Can we subtract?” 138 /2 Subtracting two-digit numbers 139

/3 Front window, rear window 141 /4 Front window, rear window-make your own 143

Num 4.10 Subtraction up to 999 /1 Race from 500 to 0 144

/2 Subtracting three-digit numbers 145 /3 Airliner 146 /4 Candy store: selling and stocktaking 148

Review Num 5.1, 5.2, and 5.3 in SAIL Volume 1Num 5.4 Number stories: abstracting number sentences /1 Number stories (multiplication) 151 /2 Abstracting number sentences 152 /3 Number stories, and predicting from number

sentences 153Num 5.5 Multiplication is commutative; alter. notations; binary mult. /1 Big Giant and Little Giant 156 /2 Little Giant explains why 158 /3 Binary multiplication 159 /4 Unpacking the parcel (binary mult.) Alter. notations 159Num 5.6 Building product tables: ready-for-use results /1 Building sets of products 162 /2 “I know another way.” 163 /3 Completing the product table (products to 45) 164 /4 Cards on the table (products to 45) 166 /5 Products practice (products to 45) 166 /6 Multiples Rummy (beginners’ pack) 168Num 5.8 Multiplying by 10 and 100 /1 Multiplying by 10 or 100 (1 die only) 176 /2 Explaining the shorthand (1 digit only) 178Review Num 6.1 and 6.2 in SAIL Volume 1Num 6.3 Division as a mathematical operation /1 Different questions, same answer. Why? 187 /2 Combining the number sentences 190 /3 Unpacking the parcel (division) 190 /4 Mr. Taylor’s Game 191Num 7.1 Making equal parts (Fifth-parts are new) /1 Making equal parts 219 /2 Same kind, different shapes 221 /3 Parts and bits 223 /4 Sorting parts 223 /5 Match and mix: parts 225

Activities Volume 2 page Activities Volume 2 page

471

Num 7.2 Take a number of like parts /1 Feeding the animals 227 /2 Traineekeepers,qualifiedkeepers 229 /3 Head keepers 230Num 7.3 Fractions as a double operation; notation /1 Expanding the diagram 233 /2 “Please may I have?” (Diagrams and notation) 235Review Space 1.5, 1.6, and 1.7 in SAIL Volume 1Space 1.8 Parallel lines, perpendicular lines /1 “My rods are parallel/perpendicular” 263 /2 “All put rods parallel/perpendicular to the big rod” 264 /3 Colouring pictures 264Space 2.1 Reflections of two-dimensional figures /1 Animals through the looking glass 314 /2 Animals two by two 314 /3 Drawing mirror images 315Space 2.2 Reflection and line symmetry /1 Introduction to symmetry 317 /2 Collecting symmetries 318NuSp 1.7 Unit intervals: the number line /1 Drawing the number line 359 /2 Sequences on the number line 360

/3 Where must the frog land? 360 /4 Hopping backwards 360 /5 Taking 360 /6 A race through a maze 361

NuSp 1.8 Extrapolation of the number line . . . the counting numbers. /1 What number is this? (Single starter) 363 /2 What number is this? (Double starter) 365 /3 Is there a limit? 366 /4 Can you think of . . .? 366NuSp 1.9 Interpolation between points. Fractional numbers (decimal) /1 What number is this? (Decimal fractions) 369 /2 Snail race 370 /3 Snails and frogs 371NuSp 1.10 Extrapolation of place-value notation /1 “How can we write this number?” (hdd. cols., stage a) 373 /2 Introducing the decimal point (tenths only) 374 /3 Pointing and writing (stage a) 375 /4 Shrinking and growing (stage a) 376Meas 1.4 International units: metre, centimetre /1 The need for standard units 385 /2 Counting centimetres with a ruler 386 /3 Mountain road 387 /4 Decorating the classroom 389Meas 1.5 Combining lengths corresponds to adding numbers of units /1 Model bridges 390 /2 How long will the frieze be? 391Meas 1.6 Different sized units for different jobs: km, mm, dm /6 Chairs in a row (dm) 396Meas 3.5 Measuring volume & capacity /non-standard units /1 Putting containers in order of capacity 433 /2 “Hard to tell without measuring” 433Meas 3.6 Standard units (litres, millilitres, and kilolitres) /1 Can you spot a litre? 434 /2 “Special: Small Lemonade, 10¢” 435 /3 Making and tasting (accuracy in the kitchen) 435 /4 Race to a litre 436Meas 4.4 Measuring mass by weighing... /1 Problem: to put these objects in order of mass 438 /2 Honest Hetty and Friendly Fred 439Meas 4.5 Standard units (kilograms) /1 Making a set of kilogram masses 441 /2 Mailing parcels 442Meas 4.6 The spring balance /1 Checking a spring balance 443 /2 Mailing parcels (spring balance) 443Meas 4.7 Grams[, tonnes] /1 “I estimate __ grams.” 444 /2 “How many grams in a litre? It all depends.” 445 /3 A portion of candy 445Review Meas 5.4 and 5.5 from SAIL Volume 1Meas 5.6 Locations in time: dates /1 What the calendar tells us 449 /2 “How long is it...?” (Same month) 450 /3 “How long is it...?” (Different month) 450 /4 “How long is it...?” (Different year) 451Meas 5.7 Locations in time: times of the day /1 “How do we know when to...?” 452 /2 “Quelle heure est-il?” 453 /3 Hours, halves, and quarters (extend to 5 minutes) 454 /4 Time, place, occupation 454Meas 6.2 Measuring temperature by using a thermometer /1 The need for a way of measuring temperature 457 /2 Using a thermometer 458

472

473

Review Num 1.11, 1.12 Org 1.13 Base ten /1 Tens and hundreds of cubes 67 /2 Tens and hundreds of milk straws 68 /3 Thousands 68Num 2.8 /2 Number targets beyond 100 87Num 2.10 Place-value notation /1 “We don’t need headings any more.” 90 /2 Number targets using place-value notation 91

/3 Place-value bingo 91Num 2.11 Canonical form /1 Cashier giving fewest coins (stage a) 95 /2 “How would you like it?” 97Num 2.12 The effects of zero /1 “Same number, or different?” (stage a) 99

/2 Less then, greater than (stage a) 100Num 2.13 Numerals beyond 100, written and spoken /1 Big numbers (to ten-thousands) 102

/2 Naming big numbers (to ten-thousands) 104Return to: Num 2.11/1, stage b) and Num 2.12/1, 2, stage b)Num 2.14 Rounding (whole numbers) /1 Run for shelter 107

/2 Rounding to the nearest hundred or thousand 109Review Meas 4.5 and 4.6 as necessaryMeas 4.7 Grams[, tonnes] /1 “I estimate__ grams.” 444 /2 “How many grams in a litre? It all depends.” 445 /3 A portion of candy 445Review Num 3.9 as necessary Num 3.10 Adding, results beyond 100 [extend to 4-digit whole nos.] /1 Start, Action, Result beyond 100 121 /2 Cycle camping 121

/3 One tonne van drivers 122 /4 Catalogue shopping 124

Review Num 4.8 and 4.9 as necessaryNum 4.10 Subtraction up to 999 [extend to 4-digit whole nos.] /1 Race from 500 to 0 144 /2 Subtracting three-digit numbers 145

/3 Airliner 146 /4 Candy store 148

Review Num 5.4 as necessaryNum 5.5 Multiplication is commutative; alter. notations; binary mult. /1 Big Giant and Little Giant 156 /2 Little Giant explains why 158 /3 Binary multiplication 159 /4 Unpacking the parcel (binary mult.) Alt. notations 159Num 5.6 Building product tables: ready-for-use results /1 Building sets of products 162 /2 “I know another way.” 163 /3 Completing the product table [products to 81] 164 /4 Cards on the table [products to 81] 166 /5 Products practice [include Variation] 166 /6 Multiples rummy [include advanced pack] 168Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers /1 Using multiplication facts for larger numbers 171 /2 Multiplying 3-digit numbers 172 /3 Cargo boats 172Num 5.8 Multiplying by 10 and 100 /1 Multiplying by 10 or 100 176 /2 Explaining the shorthand 178 Num 6.3 Division as a mathematical operation /1 Different questions, same answer. Why? 187 /2 Combining the number sentences 190 /3 Unpacking the parcel (division) 190 /4 Mr. Taylor’s Game 191Num 6.4 Organizing into rectangles /1 Constructing rectangular numbers 194 /2 The rectangular numbers game 195Num 6.5 Factoring: composite numbers and prime numbers /1 Factors bingo 197 /2 Factors rummy 197 /3 Alias prime 198 /4 The Sieve of Eratosthenes 199Num 6.6 Relation between multiplication and division /1 Parcels within parcels 201Num 6.7 Using multiplication results for division /1 A new use for the multiplication square 204 /2 Quotients and Remainders 205 /3 VillagePostOffice 207Num 6.8 Dividing larger numbers [2-digits by one digit] /1 “I’m thinking in hundreds . . .” [. . . tens] 209 /2 “I’ll take over your remainder.” 210 /3 Q and R ladders 213Review Num 7.1 and 7.2

Activities Volume 2 page

A Sequencing guide for grade Level 4 Activities Volume 2 pageNum 7.3 Fractions as a double operation; notation /1 Expanding the diagram 233 /2 “Please may I have?” (Diagrams and notation) 235Num 7.4 Simple equivalent fractions /1 “Will this do instead?” 237 /2 Sorting equivalent fractions 238 /3 Match and mix: equivalent fractions 239NuSp 1.8 Extrapolation of the number line . . . of the counting numbers. /1 What number is this? (Single starter) 363 /2 What number is this? (Double starter) 365 /3 Is there a limit? 366 /4 Can you think of . . .? 366NuSp 1.9 Interpolation between points. Fractional numbers (decimal) /1 What number is this? (Decimal fractions) 369 /2 Snail race 370 /3 Snails and frogs 371NuSp 1.10 Extrapolation of place-value notation /1 “How can we write this number?” (headed columns) 373 /2 Introducing the decimal point 374 /3 Pointing and writing (stages a and b) 375 /4 Shrinking and growing (to hundredths only) 376Review Meas 1.4 and Meas 1.6/6 as necessaryNum 7.5 Decimal fractions and equivalents /1 Making jewellery to order 240 /2 Equivalent fraction diagrams (decimal) 242 /3 Pair, and explain 243 /4 Match and mix: equivalent decimal fractions 244Num 7.6 Decimal fractions in place-value notation /1 Reading headed columns in two ways 245 /2 Same number, or different? 247

/3 Claiming and naming 248Num 7.7 Fractions as numbers. Addn. of decimal fractions in PV notation /1 Target, 1 251 /2 “How do we know that our method is still correct?” 252Review Num 2.14 (rounding to the nearest thousand)Num 7.9 Rounding decimal fractions in place-value notation /1 Rounding decimal fractions (nearest whole number only) 261Space 1.16 Classification of geometric solids /1 What must it have to be . . .? 296Space 1.19 Drawing nets of geometric solids /1 Drawing nets of geometric solids 309 /2 Recognizing solids from their nets 311 /3 Geometric nets in the supermarket and elsewhere 312Space 2.1 Reflections of two-dimensional figures /1 Animals through the looking glass 314 /2 Animals two by two 314 /3 Drawing mirror images 315Space 2.2 Reflection and line symmetry /1 Introduction to symmetry 317 /2 Collecting symmetries 318Space 2.3 Two kinds of movement: translation and rotation /1 Walking to school 320Meas 1.5 Combining lengths corresponds to adding numbers of units /1 Model bridges 390 /2 How long will the frieze be? 391Meas 1.6 Different sized units for different jobs: km, mm, dm /1 “Please be more exact.” (Telephone shopping) 392 /2 Blind picture puzzle 393 /3 “That is too exact.” (Power lines) 394 /4 Using the short lengths (Power lines) 395 /5 “That is too exact.” (Car rental) 395 /6 Chairs in a row (dm) 396Meas 2.1 Measuring area /1 “Will it, won’t it?” 405 /2 Measuring by counting tiles 406 /3 Advantages and disadvantages 406 /4 Instant tiling 407Meas 3.6 Standard units (litres, millilitres, and kilolitres) /1 Can you spot a litre? 434 /2 “Special: Small Lemonade, 10¢” 435 /3 Making and tasting (accuracy in the kitchen) 435 /4 Race to a litre 436Meas 5.7 Locations in time: times of the day /1 “How do we know when to...?” 452 /2 “Quelle heure est-il?” 453 /3 Hours, halves, and quarters (extend to minutes) 454 /4 Time, place, occupation 454Meas 6.2 Measuring temperature by using a thermometer /1 The need for a way of measuring temperature 457 /2 Using a thermometer 458Meas 6.3 Temperature in relation to experience /1 Everyday temperatures 460 /2 “What temperature would you expect?” 461 /3 Temperature in our experience 462

473

474

475

A Sequencing guide for grade Level 5 Activities Volume 2 page Activities Volume 2 page

475

NuSp 1.10 Extrapolation of place-value notation /1 “How can we write this number?” (headed columns) 373 /2 Introducing the decimal point 374 /3 Pointing and writing 375 /4 Shrinking and growing (to thousandths) 376Review Meas 1.4 and Meas 1.6/6 as necessaryNum 7.5 Decimal fractions and equivalents /1 Making jewellery to order 240 /2 Equivalent fraction diagrams (decimal) 242 /3 Pair, and explain 243 /4 Match and mix: equivalent decimal fractions 244Num 7.6 Decimal fractions in place-value notation /1 Reading headed columns in two ways 245 /2 Same number, or different? 247 /3 Claiming and naming 248Num 7.7 Fractions as numbers. Addn of decimal fractions in PV notation /1 Target, 1 251 /2 “How do we know that our method is still correct?” 252Num 7.8 Fractions as quotients /1 Fractions for sharing 255 /2 Predict, then press 257 /3 “Are calculators clever?” 258 /4 Number targets by calculator 259Review Num 2.14 (rounding to the nearest ten thousand)Num 7.9 Rounding decimal fractions in place-value notation /1 Rounding decimal fractions (nearest hundredth only) 261Space 1.10 Comparison of angles /1 “All make an angle like mine.” 270 /2 “Which angle is bigger?” 271 /3 Largest angle takes all 271 /4 Angles in the environment 272Space 1.12 Classification of polygons /1 Classifying polygons 278 /2 Polygon dominoes 279 /3 Match and mix: polygons 279Space 1.15 Classification of quadrilaterals /1 Classifying quadrilaterals 292 /2 Relations between quadrilaterals 293 /3 “And what else is this?” 294 /4 “I think you mean . . .” 294Space 1.17 Inter-relations of plane shapes /1 Triangles and polygons 298 /2 Circles and polygons 299 /3 “I can see . . .” 301 /4 Triangles and larger shapes 302Space 1.18 Tessellations /1 Tessellating regular polygons 304 /2 Tessellating other shapes 305 /3 Inventing tesselations 306 /4 Tessellating any quadrilateral 307Space 1.16 Classification of geometric solids /1 What must it have to be . . .? 296Space 1.19 Drawing nets of geometric solids /1 Drawing nets of geometric solids 309 /2 Recognizing solids from their nets 311 /3 Geometric nets in the supermarket and elsewhere 312Review Space 2.1 and 2.2 as necessary Space 2.3 Two kinds of movement: translation and rotation /1 Walking to school 320Space 2.4 Lines, rays, line segments /1 Talk like a Mathematician (lines, rays, segments) 322 /2 True or false?Space 2.5 Translationsoftwo-dimensionalfigures(slideswithoutrotation) /1 Sliding home in Flatland 326 /2 Constructing the results of slides 329 /3 Parallels by sliding 330Review Meas 1.6 Meas 1.7 Simple conversions /1 Equivalent measures, cm and mm 397 /2 Buyer beware 398Review Meas 2.1 Meas 2.2 Irregular shapes which do not fit the grid /1 Shapes and sizes 408 /2 “Hard to know until we measure” 408 /3 Gold rush 409Review Meas 3.6 Standard units (litres, millilitres, kilolitres) /5 Completing the table of units of capacity or volume 436Meas 5.7 Locations in time: times of the day /3 Hours, halves, and quarters (extend to seconds) 454 /4 Time, place, occupation 454Review Meas 6.2 Meas 6.3 Temperature in relation to experience /1 Everyday temperatures 460 /2 “What temperature would you expect?” 461 /3 Temperature in our experience 462

Num 1.13 Rectangular numbers /1 Constructing rectangular numbers (Num 6.4/1) 74

/2 The rectangular numbers game (Num 6.4/2) 75Num 1.14 Primes /1 Alias prime (Num 6.5/3) 76 /2 The sieve of Eratosthenes (Num 6.5/4) 77 /3 Sum of two primes 77Num 1.15 Square numbers /1 Square numbers 79 /2 An odd property of square numbers 79Num 1.16 Relations between numbers /1 “Tell us something new.” 82 /2 “How are these related?” 83Org 1.13 Base ten /3 Thousands 68Num 2.13 Numerals beyond 100, written and spoken /1 Big numbers (to hundred-thousands) 102

/2 Naming big numbers (to hundred-thousands) 104Return to: Num 2.11/1, stage b) and Num 2.12/1, 2, stage b)Num 2.14 Rounding (whole numbers) /1 Run for shelter 107

/2 Rounding to the nearest hundred or thousand 109 /3 Rounding big numbers (to the nearest ten thousand) 111Review Meas 4.5 and Meas 4.6 Meas 4.7 Grams, tonnes /1 “I estimate__ grams.” 444 /2 “How many grams to a litre? It all depends.” 445 /3 A portion of candy 445 /4 The largest animal ever 446Num 3.10 Adding, results beyond 100 [extend to 5-digit whole nos.] /1 Start, Action, Result beyond 100 121 /2 Cycle camping 121

/3 One tonne van drivers 122 /4 Catalogue shopping 124

Num 4.10 Subtraction up to 999 [extend to 5-digit whole nos.] /1 Race from 500 to 0 144

/2 Subtracting three-digit numbers 145 /3 Airliner 146

Review Num 5.5 as necessaryNum 5.6 Building product tables: ready-for-use results /1 Building sets of products 162 /2 “I know another way.” 163 /3 Completing the product table 164 /4 Cards on the table 166 /5 Products practice 166 /6 Multiples rummy 168Num 5.7 Multiplying 2- or 3-digit numbers by single digit numbers /1 Using multiplication facts for larger numbers 171 /2 Multiplying 3-digit numbers 172 /3 Cargo boats 172Num 5.8 Multiplying by 10 and 100 /1 Multiplying by 10 or 100 176 /2 Explaining the shorthand 178 /3 Multiplying by hundreds and thousands 178Num 5.9 Multiplying by 20 to 90 and by 200 to 900 /1 “How many cubes in this brick?” (Alternative paths) 179 /2 Multiplying by n-ty and any hundred 180Num 5.10 Long multiplication /1 Long multiplication 182 /2 Treasure chest 183Num 6.5 Factoring: composite numbers and prime numbers /1 Factors bingo 197 /2 Factors rummy 197Num 6.6 Relation between multiplication and division /1 Parcels within parcels 201 Num 6.7 Using multiplication results for division /1 A new use for the multiplication square 204 /2 Quotients and Remainders 205 /3 VillagePostOffice 207Num 6.8 Dividing larger numbers [limit: 4-digits by one digit] /1 “I’m thinking in hundreds . . .” 209 /2 “I’ll take over your remainder.” 210 /3 Q and R ladders 213 /4 Cargo Airships 214Num 7.3 Fractions as a double operation; notation /1 Expanding the diagram 233 /2 “Please may I have?” (Diagrams and notation) 235Num 7.4 Simple equivalent fractions /1 “Will this do instead?” 237

/2 Sorting equivalent fractions 238 /3 Match and mix: equivalent fractions 239Review NuSp 1.8 and 1.9 as necessary

476

477 477

A Sequencing guide for grade Level 6

Num 1.14 Primes /1 Alias prime (Num 6.5/3) 76 /2 The sieve of Eratosthenes (Num 6.5/4) 77 /3 Sum of two primes 77Num 1.15 Square numbers /1 Square numbers 79 /2 An odd property of square numbers 79Num 1.16 Relations between numbers /1 “Tell us something new.” 82 /2 “How are these related?” 83Org 1.13 Base ten /3 Thousands 68Num 2.13 Numerals beyond 100, written and spoken /1 Big numbers (to billions) 102

/2 Naming big numbers (to billions) 104Return to: Num 2.11/1, stage b) and Num 2.12/1, 2, stage b)Num 2.14 Rounding (whole numbers) /1 Run for shelter 107

/2 Rounding to the nearest hundred or thousand 109 /3 Rounding big numbers (nearest hundred million) 111Meas 4.7 Grams, tonnes /1 “I estimate __ grams.” 444 /2 “How many grams in a litre? It all depends.” 445 /3 A portion of candy 445 /4 The largest animal ever 446Review Num 3.10 as necessaryReview Num 4.10 as necessaryReview Num 5.6 as necessaryNum 5.7 Multiplying 2- or 3-digit numbers by single digit numbers /1 Using multiplication facts for larger numbers 171 /2 Multiplying 3-digit numbers 172 /3 Cargo boats 172Num 5.8 Multiplying by 10 and 100 /1 Multiplying by 10 or 100 176 /2 Explaining the shorthand 178 /3 Multiplying by hundreds and thousands 178Num 5.9 Multiplying by 20 to 90 and by 200 to 900 /1 “How many cubes in this brick?” (Alternative paths) 179 /2 Multiplying by n-ty and any hundred 180Num 5.10 Long multiplication /1 Long multiplication 182 /2 Treasure chest 183Review Num 6.5, 6.6 and 6.7 as necessaryNum 6.8 Dividing larger numbers [limit: 4-digits by two digits] /1 “I’m thinking in hundreds . . .” 209 /2 “I’ll take over your remainder.” 210 /3 Q and R ladders 213 /4 Cargo Airships 214Num 6.9 Dvision by calculator /1 Number targets: division by calculator 217Review Num 7.3 and 7.4 as necessaryReview NuSp 1.8, 1.9 and 1.10 as necessaryReview Num 7.5 and 7.6 as necessaryNum 7.7 Fractions as numbers. Addn of decimal fractions in PV notation /1 Target, 1 251 /2 “How do we know that our method is still correct?” 252Num 7.8 Fractions as quotients /1 Fractions for sharing 255 /2 Predict, then press 257 /3 “Are calculators clever?” 258 /4 Number targets by calculator 259Review Num 2.14 (rounding to the nearest hundred million)Num 7.9 Rounding decimal fractions inplace value notation /1 Rounding decimal fractions (nearest hundredth only) 261 /2 Fractional number targets 262Space 1.9 Circles /1 Circles in the environment 266 /2 Parts of a circle 266 /3 Circles and their parts in the environment 267 /4 Patterns with circles 268Space 1.10 Comparison of angles /1 “All make an angle like mine.” 270 /2 “Which angle is bigger?” 271 /3 Largest angle takes all 271 /4 Angles in the environment 272Space 1.11 Classification of angles /1 Right angles, acute angles, obtuse angles 273 /2 Angle dominoes 274 /3 “Mine is the different kind.” 275 /4 “Can’tcross,willfit,mustcross.” 275Space 1.12 Classification of polygons /1 Classifying polygons 278 /2 Polygon dominoes 279 /3 Match and mix: polygons 279

Activities Volume 2 page Activities Volume 2 pageSpace 1.15 Classification of quadrilaterals /1 Classifying quadrilaterals 292 /2 Relations between quadrilaterals 293 /3 “And what else is this?” 294 /4 “I think you mean . . .” 294Space 1.17 Inter-relations of plane shapes /1 Triangles and polygons 298 /2 Circles and polygons 299 /3 “I can see . . .” 301 /4 Triangles and larger shapes 302Space 1.18 Tessellations /1 Tessellating regular polygons 304 /2 Tessellating other shapes 305 /3 Inventing tesselations 306 /4 Tessellating any quadrilateral 307Space 1.16 Classification of geometric solids /1 What must it have to be . . .? 296Space 1.19 Drawing nets of geometric solids /1 Drawing nets of geometric solids 309 /2 Recognizing solids from their nets 311 /3 Geometric nets in the supermarket and elsewhere 312Space 1.13 Polygons: Congruence & similarity [prep. for ratio - implict] /1 Congruent and similar polygons 283 /2 Sides of similar polygons 284 /3 Calculating lengths from similarities 285Space 1.14 Triangles: Classification, congruence, similarity /1 Classifying triangles 287 /2 Triangle dominoes 288 /3 Match and mix: triangles 288 /4 Congruent and similar triangles 289Review Space 2.1 and 2.2 as necessary Space 2.3 Two kinds of movement: translation and rotation /1 Walking to school 320Space 2.4 Lines, rays, line segments /1 Talk like a Mathematician (lines, rays, segments) 322 /2 True or false? 323Space 2.5 Translations of two-dimensional figures (slides without rotation) /1 Sliding home in Flatland 326 /2 Constructing the results of slides 329 /3 Parallels by sliding 330Space 2.6 Directions in space: north, south, east, west, and the half points /1 Compass directions 332 /2 Directions for words 333 /3 Island cruising 334Space 2.7 Angles as amount of turn; compass bearings /1 Directions and angles 338 /2 Different name, same angle 339 /3 Words from compass bearings 339 /4 Escape to freedom 341Space 2.8 Directions and locations /1 Introduction to back bearings 344 /2 Get back safely 345 /3 Where are we? 347 /4 True north and magnetic north 350Space 2.9 Rotations of two-dimensional figures /1 Centre and angle of rotation 352 /2 Walking the planks 353 /3 Introduction to rotational symmetry 355 /4 Match and mix: rotational symmetries 356 /5 Match and mix (line symmetry) 357Space 2.10 Relation between reflections, rotations, and flips 358Meas 1.7 Simple conversions /1 Equivalent measures, cm and mm 397 /2 Buyer beware 398 /3 A computer-controlled train 399 Meas 1.8 The system overall /1 Relating different units 402 /2 “Please may I have?” (Metre and related units) 403Review Meas 2.1 and 2.2 Meas 2.3 Rectangles (whole number dimensions); meas. by calculation /1 “I know a shortcut.” 411 /2 Claim and explain 412 /3 Carpeting with remnants 412Meas 2.4 Other shapes made up of rectangles /1 ‘Home improvement’ in a doll’s house 414

/2 Claim and explain (harder) 415 /3 Net of a box 415 /4 Tilingthefloorsinahome 416EXTENSION: MEAS 2.5, 2.6, 2.7 and 2.8Review Meas 3.6/5 436Meas 4.8 /1 Completing the table of units of mass 447Review Meas 5.7 (extend to 24-hour clock) Meas 5.8 Equivalent measures of time /1 Pairing equivalent times 456Review Meas 6.2 and 6.3

478

479

13

11 12 13 14 15 16

Structured Activities for Intelligent Learning Progress Record Name _____________________________ Grade Level 3 Grade Level 4 Grade Level 5 Grade Level 6

8 (13)

Org 1

Num 1

Num 2

Num 3

Num 4

Num 5

Num 6

Num 7

Space 1

Space 2

NuSp 1

Patt 1

Meas 1

Meas 2

Meas 3

Meas 4

Meas 5

Meas 6

4

7

9 10 11 12 13 14 (13) 14 (13) (14)

9 10 (10) (10)

(10)(10)109876

4 5 6 (8) 7 8 8 9 10

3 4 5 6 7 8 8 9

1 2 3 4 5 6 7 (9) 8 9 9

8 16 19 10 12 15 17 18 9 11 13 14

2 3 4 5 6 7 8 9 101

7 8 9 10

4

4 5 6 7 7 8(6)

1 2 3 5 6 7 8

6 65

4 5 6 7 8

8(7)(7)76

2 3